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THE 


UNIVEESITY  ARITHMETIC, 


BMBRACING  THE 


SCIENCE  OF  NUMBERS, 


AND    THEIR    NUMEROUS    APPLICATIONS 


B¥ 


CHARLES  DAVIES,  LL.  1)., 

A.UTHOR  OF  FIRST  LESSONS  IN  ARITHMETIC  ;   ARITHMETIC  ;   ELEMENTARY  ALGEBRA  ; 
ELEMENTARY   GEOMETRY;    ELEMENTS   OF   DRAWING   AND   MENSURATION; 
ELEMENTS   OF   SURVEYING  ;    ELEMENTS   OF  ANALYTICAL  GEOM- 
ETRY ;     DESCRIPTIVE    GEOMETRY  ;     SHADES,    SHADOVi^S, 
AND  PERSPECTIVE  ;   AND  DIFFERENTIAL  AND 
INTEGRAL    CALCULUS 


BEVISED     AND     IMPROVED     EDITION. 

NEW   YORK: 
PUBLISHED    BY   A.  S.   BARNES   &   CO., 

No     51   JOHN    STREET. 

1852. 


Entered  according  to  Act  of  Congress,  in  the  year  Eighteen  Hundred  and  Fifty 

BY    CHARLES    DAVIES, 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the  Soutnern  District 
of  New- York. 


PREFACE. 


Science,  in  its  popular  signification,  means  knowledge 
reduced  to  order;  that  is,  knowledge  so  classified  and 
arranged,  as  to  be  easily  remembered,  readily  referred 
to,  and  advantageously  applied. 

Arithmetic  is  the  science  of  numbers.  It  lies  at  the 
foundation  of  the  exact  and  mixed  sciences,  and  a  know- 
ledge of  it  is  an  important  element  either  of  a  liberal  or 
practical  education.  While  Arithmetic  is  a  science  in 
all  that  concerns  the  properties  of  numbers,  it  is  yet  an 
art  in  all  that  relates  to  their  practical  application.  It  is 
the  first  subject  in  a  well-arranged  course  of  instruction 
to  which  the  reasoning  powers  of  the  mind  are  applied, 
and  is  the  guide-book  of  the  mechanic  and  man  of  busi- 
ness. It  is  the  first  fountain  at  which  the  young  votary 
of  knowledge  drinks  the  pure  waters  of  intellectual  truth. 

It  has  seemed  to  the  author  of  the  first  importance  that 
this  subject  should  be  well  treated  in  our  Elementary 
Text  Books.  In  the  hope  of  contributing  something  to 
so  desirable  an  end,  he  has  prepared  a  series  of  arithmeti- 
cal works,  embracing  three  books,  entitled  First  Lessons 
m  Arithmetic  ;  Arithmetic  ;  and  University  Arithmetic — 
the  latter  of  which  is  the  present  volume 

The  First  Lessons  in  Arithmetic  are  designed  for  be- 
ginners. The  subjects  treated  are  divided  into  separate 
lessons,  each  lesson  embracing  one  combination  of  num- 
bers, or  one  set  of  combinations. 

The  Arithmetic  is  designed  for  the  use  of  schools  and 


Vi  PREFACE 

academies,  and  contains  all  that  is  usually  taught  m  a 
course  of  academical  instruction. 

The  University  Arithmetic  is  intended  to  answer  an- 
other object.  In  it,  the  entire  subject  is  treated  as  a 
science.  The  scholar  is  supposed  to  be  familiar  with  the 
operations  in  the  four  ground  rules,  which  are  now  taught 
to  small  children  either  orally  or  from  elementary  trea- 
tises. This  being  premised,  the  language  of  figures, 
which  are  the  representatives  of  numbers,  is  carefully 
taught,  and  the  different  significations  of  which  the  fig- 
ures are  susceptible,  depending  on  the  manner  in  which 
they  are  written,  are  fully  explained.  It  is  shown,  for 
example,  that  the  simple  numbers  in  which  the  value  of 
the  unit  increases  from  right  to  left  according  to  the  scale 
of  tens,  and  the  Denominate  or  Compound  numbers  in 
which  it  increases  according  to  a  different  scale,  belong 
in  fact  to  the  same  class  of  numbers,  and  that  both  may 
be  treated  under  a  common  set  of  rules.  Hence,  the 
rules  for  Notation,  Addition,  Subtraction,  Multiplication, 
and  Division,  have  been  so  constructed  as  to  apply  equally 
to  all  numbers.  This  arrangement,  which  the  author  has 
not  seen  elsewhere,  is  deemed  an  essential  improvement 
in  the  science  of  Arithmetic. 

In  developing  the  properties  of  numbers,  from  their 
elementary  to  their  highest  combinations,  great  labor  has 
been  bestowed  in  classification  and  arrangement.  It  has 
been  a  leading  object  to  present  the  entire  subject  of 
irithmetic  as  forming  a  series  of  dependent  and  con- 
nected propositions  :  so  that  the  pupil,  while  acquiring 
useful  and  practical  knowledge,  may  at  the  same  tihne  be 
introduced  to  those  beautiful  methods  of  exact  reasonmg, 
which  science  alone  can  teach. 

Great  care  has  also  been  taken  to  demonstrate  fully  al' 


PREFACE.  Vll 

the  rules  and  to  explain  the  reason  of  every  process  from 
the  most  simple  to  the  most  difficult.  It  has  been  thought 
that  the  Teachers  of  the  country  would  like  to  possess  a 
work  of  this  kind,  and  that  it  might  be  studied  advan- 
tageously as  a  text  book  in  our  advanced  schools  and 
academies.  To  adapt  it  to  such  a  use,  a  largfe  number 
of  practical  examples  has  been  added,  many  of  which 
have  been  selected  from  an  English  work  by  Keith. 

In  the  preparation  of  the  work,  another  object  has  been 
kept  constantly  in  view,  viz.,  to  adapt  it  to  the  business 
wants  of  the  country.  For  this  purpose  much  pains 
have  been  bestowed  in  the  preparation  of  the  articles  on 
Weights  and  Measures,  foreign  and  domestic ;  on  Bank- 
ing, Bank  Discount,  Interest,  Coins  and  Currency,  and 
Exchanges. 

Although  by  law  the  hundred  weight  is  estimated  at 
100  pounds,  and  consequently  the  quarter  at  25  pounds, 
in  the  United  States,  yet  the  old  hundred  of  112  pounds 
is  still  much  used  ;  and  in  all  our  intercourse  with  Great 
Britain,  goods  and  wares  are  so  estimated.  Hence,  it 
was  thought  best  in  this  arithmetic,  intended  for  general 
instruction,  to  retain  the  old  standard. 

In  fine,  it  has  been  the  aim  of  the  author  to  publish 
both  a  scientific  and  practical  treatise  on  the  subject  of 
Arithmetic,  and  one  which  shall  in  some  measure  cor- 
respond to  the  higher  qualifications  of  teachers  and  the 
improved  methods  of  communicating  instruction. 

Several  excellent  works,  of  an  elementary  character, 
having  recently  been  published  on  Book-keeping,  it 
has  seemed  best  to  omit,  in  the  present  edition,  the  arti- 
cle on'  that  subject,  and  to  supply  its  place  by  matter 
of  a  practical  character. 

FisHKiLL  Landing.  January,   1850. 


DAVIES' 
COURSE  OF  MATHEMATICS. 


DAVIES'  FIRST  LESSONS  IN  ARITHMETIC— For  beginners. 

DA  VIES'  ARITHMETIC— Designed  for  the  use  of  Academies  and 
Schools. 

KEY  TO  DAVIES'  ARITHMETIC 

DAVIES'  UNIVERSITY  ARITHMETIC— Embracing  the  Science 
of  Numbers,  and  their  numerous  appHcations. 

KEY  TO  DAVIES'  UNIVERSITY  ARITHMETIC 

DAVIES'  ELEMENTARY  ALGEBRA— Being  an  Introduction  to 
the  Science,  and  forming  a  connecting  hnk  between  Arithmetic  and 
Algebra. 

KEY  TO  DAVIES'  ELEMENTARY  ALGEBRA. 

DAVIES'  ELEMENTARY  GEOMETRY.— This  work  embraces  the 
elementary  principles  of  Geometry.  The  reasoning  is  plain  and  con- 
cise, but  at  the  same  time  strictly  rigorous. 

DAVIES'  ELEMENTS   OF  DRAWING  AND  MENSURATION 

— Applied  to  the  Mechanic  Arts. 

DAVIES'  BOURDON'S  ALGEBRA— Including  Sturms'  Theorem,— 
Being  an  Abridgment  of  the  work  of  M.  Bourdon,  with  the  addition  of 
practical  examples. 

])AVIES'   LEGENDRE'S   GEOMETRY  and  TRIGONOMETRY. 

— Being  an  Abridgment  of  the  work  of  M.  Legendre,  with  the  addition 
of  a  Treatise  on  Mensuration  of  Planes  and  Solids,  and  a  Table  of 
Logarithms  and  Logarithmic  Sines. 

DAVIES'  SURVEYING— With  a  description  and  plates  of  the  Theod- 
olite, Compass,  Plane-Table,  and  Level  :  also.  Maps  of  the  Topo- 
graphical Signs  adopted  by  the  Engineer  Department — an  explana- 
tion of  the  method  of  surveying  the  Public  Lands,  and  an  Elementary 
Treatise  on  Navigation. 

DAVIES'  ANALYTICAL  GEOMETRY— Embracing  the  EutJA- 
tions  of  the  Point  and  Straight  Line — of  the  Conic  Sections — oi 
the  Line  and  Plane  in  Space — also,  the  discussion  of  the  General 
Equation  of  the  second  degree,  and  of  Surfaces  of  the  second  order. 

DWIES'  DESCRIPTIVE  GEOMETRY,— With  its  application  to 
kspherical  Projections. 

DAVIES'  SHADOWS  and  LINEAR  PERSPECTIVE. 

DAVIES'  DIFFERENTIAL  and  INTEGRAL  CALCULUS. 


CONTENTS. 


FIRST    FIVE    RULES. 

Page 

Notation  and  Numeration 13 — 19 

Of  the  Signs 14—15 

Of  the  Denomination  of  Numbers 19 — 20 

Tables  of  Money,  Weights,  Measures,  &c., — Americein  and 

Foreign 20—39 

Remarks  on  the  Formation  of  Numbers 40—41 

Of  Reduction 41 — 45 

Addition ^     46 — 60 

Subtraction 61 — 68 

MultipHcation 69 — 80 

Division 81—93 

Of  the  Properties  of  the  9'8... , 93—97 

Remarks 98 

Divisions  of  Arithmetic 99 

VULGAR    FRACTIONS. 

Definition  of,  and  First  Principles 100 — 103 

The  six  Kinds  of  Fractions 103—104 

Six  Propositions 105 — 109 

Greatest  Common  Divisor 109 — 114 

Second  Method  of  finding 112—114 

Least  Common  Multiple 114 — 117 

First  Method  of  finding 115—116 

Second  Method 116—117 

Reduction  of  Vulgar  Fractions 117 — 128 

Reduction  of  Denominate  Fractions 128 — 134 

Addition  of  Vulgar  Fractions 135—139 

Subtraction  of  Vulgar  Fractions 139 — 141 

Multiplication  of  Vulgar  Fractions 141 — 145 

->ivision  of  Vulgar  Fractions 145 — 148 


X  ^  CONTENTS. 

DECIMAL    FRACTIONS. 

Page 

Definition  of  Decimals,  &,c 149 — 150 

Decimal  Numeration  Table— First  Principles,  &c 1 50 — 1 54 

Addition  of  Decimals 155 — 156 

Subtraction  of  Decimals 156 — 157 

Multiplication  of  Decimals 158 — 159 

Contraction  in  Multiplication v 159 — 161 

Division  of  Decimals 1 62 — 1 65 

Applications  in  the  Four  Rules 165 — 166 

Contraction  in  Division 167 — 168 

Reduction  of  Vulgar  Fractions  to  Decimals 168 — 170 

Reduction  of  Denominate  Decimals 170 — 174 

Circulating  or  Repeating  Decimals — Definition  of,  &c 175 — 178 

Reduction  of  Circulating  Decimals 178 — 184 

Addition  of  Circulating  Decimals 184 — 185 

Subtraction  of  Circulating  Decimals 185 

Multiplication  of  Circulating  Decimals 186 

Division  of  Circulating  Decimals..... 187 

KATIO    AND    PROPORTION    OF   NUMBERS. 

Ratio  Defined  and  Illustrated J88— 189 

Proportion  Defined  and  Illustrated - .  190 — 192 

Of  Cancelling 193—195 

Rule  of  Three— Defined,  Proof,  &c 196—202 

Rule  of  Three  by  Analysis 203—204 

Rule  of  Three  by  Cancelling 205—206 

Examples  involving  Fractions 207 — 208 

Of  Questions  requiring  two  Statements , 209 — 210 

Double  Rule  of  Three — Definition,  Demonstration,  &c 210 — 214 

PRACTICE — TARE    AND    TRET. 

Practice — Definition  of,  &c 215 

Table  of  Aliquot  Parts 215 

Examples  illustrating  Principles,  &-c 216-  -219 

Tare  and  Tret 220—222 

PERCENTAGE— INTEREST. 

Percentage 223—224 

Interest — Definition  of 225 

Principles,  and  ways  of  finding  Interest 226 — 237 

Applications  237 — 238 

Partial  Payments— New  York  Rule 238—241 

Questions  in  Interest 241 — 243 


CONTENTS.  ;X1 

Page 
Table  showing  the  legal  Rate  of  Interest  in  each  State,  Num- 
ber of  Shillings  to  the  Dollar,  Value  of  the  Dollar  in 

Pounds,  &c 244 

Reduction  of  Currencies 244 — 245 

Compound  Interest — Definition  of,  &c 246 — 247 

Table  showing  the  Interest  of  £1  or  $1,  &c 247—248 

APPLICATIONS  TO  BUSINESS. 

Loss  and  Gain 249—252 

Stocks  and  Coi-porations 252 — 253 

Commission  and  Brokerage... 253 — 256 

Banking 256—257 

Forms  of  Notes 257 

Remarks  relating  to  Notes 258 — 259 

Baiik  Discount... 259—262 

Discount 262—264 

Insurance 265 — 266 

Assessing  Taxes 266 — 269 

Equation  of  Payments 269—272 

Partnership  or  Fellowship 273 — 274 

Double  Fellowship 274—276 

Alligation  Medial 276—277 

Alligation  Alternate ^ 277—282 

Custom  House  Business 282 — 285 

Forms  relating  to  Business  in  General 285 — 287 

Forms  of  Orders 285 

Forms  of  Receipts 285—286 

Forms  of  Bonds 286—287 

General  Average 288—290 

Tonnage  of  Vessels 291—292 

Custom  House  Charges  on  Vessels 291 

Government  Rule — Carpenter's  Rule 292 

Gauging— Varieties  of  Casks,  &c 293 — 296 

Life  Insurance 296—299 

Table  showing  the  Expectation  of  Life 297 

Rates  of  Insurance  on  Life 298 

Endowments  and  Annuities 299 — 300 

Coins  and  Currencies — Definition  of. 301 

Values  of  Foreign  Coins 301 — 303 

Exchange — Definition  of 303 

Bills  of  Exchange 304—305 

Endorsing  Bills 306 

Acceptance — Liabilities  of  the  Parties 306 — 307 

Par  of  Exchange — Course  of  Exchange 307 — 308 


Xii  CONTENTS. 

Page 

Examples. 309—312 

Arbitration  of  Exchange 312—314 

DUODECIMALS. 

Definition  of,  &c 315—316 

Multiplication  of  Duodecimals... 316 — 318 

INVOLUTION. 

Definition  of,  &c 318—319 

EVOLUTION. 

Definition  of,  &c 320 

Extraction  of  the  Square  Root 320— -326 

Extraction  of  the  Cube  Root 326—331 

ARITHMETICAL    PROGRESSION. 

Definition  of,  &c 331—332 

Different  Cases 333—335 

Genered  Examples 335 

GEOMETRICAL    PROGRESSION,    ETC. 

Definition  of,  &c 336—337 

Cases 337—338 

MENSURATION,    ETC. 

To  find  the  area  of  a  Triangle. 3S9— 840 

To  find  the  area  of  a  Square,  Rectangle,  &c 340 — 341 

To  find  the  area  of  a  Trapezoid 341—342 

To  find  the  circumference  and  diameter  of  a  Circle 343 — 344 

To  find  the  area  of  a  Circle 344 

To  find  the  surface  of  a  Sphere 344 — 345 

To  find  the  solidity  of  a  Sphere 346 

To  find  the  convex  surface  of  a  Prism 346 — 347 

To  find  the  solid  contents  of  a  Prism 347 

To  find  the  convex  surface  of  a  Cylinder 348 

To  find  the  sohdity  of  a  Cylinder 348—349 

To  find  the  solidity  of  a  Pyramid 849—350 

To  find  the  solidity  of  a  Cone 350—351 

Right-Angled  Triangle 351—352 

Mechanical  Powers 353—860 

Promiscuous  Questions 361 — 367 


ARITHMETIC. 


NOTATION  AND   NUMERATION. 

1.  Science  in  its  popular  sense,  is  knowledge  reduced  to 
order:  that  is,  knowledge  so  classified  and  arranged,  as  to  be 
easily  remembered,  readily  referred  to,  and  advantageously 
applied.  In  a  strictly  technical  sense,  it  refers  to  the  laws  which 
connect  the  facts  and  principles  of  any  subject  of  knowledge 
with  each  other. 

2.  Arithmetic  is  both  a  science  and  an  art.  It  is  a  science 
in  all  that  concerns  the  properties,  laws  and  proportions  of 
numbers;  and  an  art  in  all  that  relates  to  their  uses  and 
applications. 

3.  Numbers  are  expressions  for  oile  or  more  things  of 
the  same  kind :  thus,  the  words  one,  two^  three^  four^  five,  six, 
seven,  eight,  nine,  ten,  eleven,  twelve,  thirteen,  &c.,  are  called 
numbers. 

4.  The  unit  of  a  number  is  one  of  the  equal  things  which 
the  number  expresses.  Thus,  if  the  number  express  six  ap- 
ples, one  apple  is  the  unit;  if  it  express  five  pounds  of  tea, 
one  pound  of  tea  is  the  unit ;  if  ten  feet  of  length,  one  foot  is 
the  unit;  if  foiir  hours  of  time,  one  hour  is  the  unit. 

5.  In  common  language  numbers  are  expressed  by  words : 
in  the  language  of  arithmetic  they  are  generally  expressed 
by   figures.     In   our  language   there  are  twenty-six  difierent 

Quest. — 1.  What  is  Science  ?  2.  What  is  Arithmetic  ?  When  is  it  a 
science  and  when  an  art  ?  3.  What  are  numbers  ?  Give  an  example. 
4.  What  is  the  unit  of  a  number  ?  What  is  the  unit  of  six  apples  ?  Of 
five  pounds  of  tea  ?  Of  ten  feet  in  length  ?  Of  four  hours  of  time  ? 
6.  How  are  numbers  expressed  in  common  language  ?  How  are  they  ex- 
pressed in  the  language  of  arithmetic  ?  How  many  characters  are  there 
in  our  language  ? 

(     13     ) 


14  NOTATION     AND     NUMERATION. 

characters  called  letters :  in  the  language  of  arithmetic  there 
are  but  ten  characters  which  represent  numbers  ;  they  are 
called  figures.     They  are 
naught,  one,  two,  three,  four,  five,  six,  seven,  eight,  nine 
0  123456789 

The  character  0  is  used  to  denote  the  absence  of  a  thing. 
As,  if  we  wish  to  express  by  figures  that  there  are  no  apples 
in  a  basket,  we  write,  the  number  of  apples  in  the  basket 
is  0.  The  nine  other  figures  are  called  significant  figures,  or 
digits. 

6.  Besides  the  figures  which  represent  numbers,  there  are 
certain  other  characters  used,  called  signs,  which  indicate 
the  operations  to  be  performed  on  numbers.  They  are  the 
following : 

The  sign  -}-  is  called  plus,  and  when  placed  between  two 
numbers,  indicates  that  they  are  to  be  added  together :  thus, 
3  +  2  shows  that  3  and  2  are  to  be  added,  and  is  read, 
3  plus  2. 

The  sign  —  is  called  minus,  and  when  placed  between 
two  numbers,  indicates  that  the  one  on  the  right  is  to  be  taken 
from  the  one  on  the  left :  thus,  4  —  3  shows  that  3  is  to  be 
taken  from  4,  and  is  read,  4  minus  3. 

The  sign  =  is  called  the  sign  of  equality,  and  when  placed 
between  two  numbers,  indicates  that  they  are  equal  to  each 
other :  thus,  2  +  3  =  5  shows  that  2  added  to  3  gives  a  sum 
equal  to  5,  and  is  read,  2  plus  3  equals  5. 

The  sign  X  is  called  the  sign  of  multiplication,  and  when 
placed  between  two  numbers,  indicates  that  they  are  to  be 
multiplied  together :  thus,  12x3  shows  that  12  is  to  be 
multiplied  by  3,  and  is  read,  12  multiplied  by  3. 

The  sign  -^  is  called  the  sign  of  division,  and  when  placed 
between  two  numbers,  indicates  that  the  one  on  the  left  is  to 

Quest. — ^Wliat  are  the  characters  called?  In  arithmetic,  how  many 
characters  are  there  which  represent  numbers?  What  are  they  called? 
Name  them.  What  is  the  0  used  for  ?  What  are  the  other  nine  figures 
called?  6.  What  signs  are  used  to  indicate  the  operations  to  be  performed 
on  numbers?     Name  each,  and  explain  its  use. 


NOTATION     AND     NUMERATION. 


15 


three 

U                       CI 

four 

U                       (( 

five 

((                       tl 

six 

((                            C( 

seven 

((                 (( 

eight 

H                           ii 

by  2, 

by  3, 

by  4, 

by  5, 

by  6, 

by  7, 

by?, 

by  9. 

be  divided  by  the  one  on  the  right :  thus,  8  -f-  4  shows  that  8 
is  to  be  divided  by  4  :   and  is  read,  8  divided  by  4. 

The  parenthesis  is  used  to  indicate  that  the  sum  of  two 
or  more  separate  numbers  is  to  be  muUiplied  by  a  single 
Qumber :  thus,  (3  -f  5)  x  6  shows  that  the  sum  of  3  and  5 
s  to  be  muUiplied  by  6. 

7.  We  have  now  learned  the  alphabet  of  the  arithmeti- 
cal language,  and  understand  that 

A  single  thing,  or  a  unit  of  a  number,  may  be  expressed  by  1, 
two  things  of  the  same  kind,  or  two  units, 

or  three  units 

or  four  units 

or  five  units 

or  six  units 

or  seven  units  * 

or  eight  units 

or  nine  units 

The  units  of  the  numbers  expressed  above  are  called  sim- 
ple units,  or  units  of  the  first  order. 

8.  The  next  step,  in  the  arithmetical  language,  is  to  write 
the  0  on  the  right  of  the  1  ;  thus,  10.  This  sign  is  the  arith- 
metical expression  for  the  word  ten.  The  character  1  still 
expresses  a  single  thing,  viz.,  one  ten.  This  ten,  however, 
is  ten  times  as  great  as  the  simple  unit,  and  is  called  a  unit 
of  the  second  order. 

9.  We  next  write  two  O's  on  the  right  of  the  1  ;  thus, 
100.  This  is  the  arithmetical  expression  for  one  hundred, 
that  is,  for  ten  tens.  Here,  again,  the  1  expresses  but  a  sin- 
gle thing,  viz.,  one  hundred ;  but  this  one  hundred  is  equal 
to  ten  units  of  the  second  order,  or  to  one  hundred  units  of 
the  first  order.     In  a  similar  manner  we  may  form  as  many 

Quest. — 7.  What  character  stands  for  four  things?  What  for  eight! 
What  are  the  units  of  such  numbers  called?  8.  What  is  the  next  step 
in  tlie  language  of  figiures  ?  What  does  1  still  express  ?  What  is  the  single 
thing  called?  What  is  it  equal  to?  9.  What  is  the  next  step?  What  does 
1  still  express  ?  To  how  many  units  of  the  second  order  is  it  equal  ?  To 
how  many  of  the  first  ? 


16  NOTATION  AND  NUMERATION. 

orders  of  units  as  we  please :  thus,  a  single  unit  of  the  first 

order  is  expressed  by 1, 

a  unit  of  the  second  order  by  1  and  a  0  ;  thus,  10, 

a  unit  of  the  third  order      by  1  and  two  O's  ;  thus,  100, 

a  unit  of  the  fourth  order  by  1  and  three  O's  ;  thus,  1000, 
a  unit  of  the  fifth  order  by  1  and  four  O's  ;  thus,  10000, 
a  unit  of  the  sixth  order  by  1  and  five  O's  ;  thus,  100000, 
and  so  on  for  the  units  of  higher  orders. 

When  units  simply  are  named,  units  of  the  first  order  are 
always  meant. 

10.  We  see,  from  the  language  of  figures,  that  units  of 
the  first  order  always  occupy  the  place  on  the  right ;  units 
of  the  second  order  the  second  place  from  the  right ;  units 
of  the  third  order,  the  third  place ;  and  so  on  for  places  still 
to  the  left. 

We  also  see  that  ten  units  of  the  first  order  make  one  of 
the  second ;  ten  of  the  second,  one  of  the  third ;  ten  of  the 
third,  one  of  the  fourth ;  and  so  on  for  the  higher  orders. 
Hence,  the  language  expresses  that,  When  figures  are  written 
hy  the  side  of  each  other,  ten  units  of  any  one  place  make  one 
unit  of  the  place  next  to  the  left. 

11.  For  the  purpose  of  reading  figures,  they  are  often 
separated  into  periods  of  three  figures  each.  The  units  of 
the  first  order  are  read,  simply,  units ;  those  of  the  second 
order  are  generally  read,  tens ;  those  of  the  third,  hundreds ; 
those  of  the  fourth,  thousands,  &c.,  according  to  the  follow- 
ing 

Quest. — How  is  a  single  unit  of  the  first  order  expressed?  How  do 
5'^ou  express  one  unit  of  the  second  order  ?  One  of  the  third  ?  One  of  the 
fourth  ?  One  of  the  fifth  ?  10.  What  places  do  units  of  different  orders  oc- 
cupy ?  When  figures  are  written  by  the  side  of  each  other,  how  many 
units  of  one  order  make  one  unit  of  the  place  next  to  the  left?  11.  How 
are  figures  separated  for  the  purpose  of  reading?  How  are  units  of  the 
first  order  read  ?  Those  of  the  second  ?  Those  of  the  third  ?  Those  of  the 
fourth,  &c.? 


NOTATION     AND     NUMERATION.  ^7 

NUMERATION  TABLE.* 


« 

'TD 

th  period, 
r  period 
f  Quintillic 

th  period, 
r  period 
f  Quadrilli( 

1  period, 

period 

Trillions, 

1  period, 
)eriod 
3illions. 

.2  o 

03     q1 

.2 

period, 

period 

Thousan 

t  period, 
period 
Units. 

*^  ^ 

«*-! 

-!->        U 

l+H 

■XJ     M 

(4-4 

■T3     H  «*-. 

Oi      f^    ^ 

i>  o  o 

50    0    0 

o  o 

O 

-^  o 

O 

CO    o 

0 

c^  0  0 

^irA^ 

1 

.2 

02 

■^   eo 

09 

S2 

09 

§ 

1     S 

.2  o 

3  15 

•;§§ 

.2 

g 

.2 

§^ 

§■=1   ■ 

a  « 

1=5     02 

2  fl 

o^-^ 

a-^ 

H.2 

mg 

§1 

L-i    w 

v-.S 

«*-•    c3    ,« 

«*-  "^ 

«^  li::^ 

«+-<  :id 

s^    0 

reds  0 
of  Qu: 
illions 

^4  = 
M  C7   O 

03  PQ 

o  r;^ 

|h^ 

09 

1o| 

03 

o 

1*3 

C 

J-o 

03 

I'sl 

n3 

03 

2  G  .5 

^2    CO  ^3 

i=5    C    c^ 

1  = 

1- 

o 

|S 

0 

^2    M    3 

Ceo 

i  i.« 

S    o    :3 

^    03    3 

S3    03 

P    q; 

3    03 

3    03  -G 

.^    03    H 

WHO? 

Wna 

WHH 

WHW 

WH^ 

WHH 

wh^ 

The  words  at  the  head  of  the  numeration  table,  units, 
tens,  hundreds,  &c.,  are  equally  applicable  to  all  numbers, 
and  must  be  committed  to  memory.  The  table  may  be  con- 
tinued to  any  extent.  The  higher  periods  take  the  names  of 
Sextillions,  Septillions,  Octillions,  Nonillions,  Decillions,  Un- 
decillions,  Duodecillions. 

12.  Expressing  or  writing  numbers  in  figures  is  called 
NOTATION  Reading  the  signification  of  the  figures  correctly, 
when  written,  is  called  numeration. 

EXAMPLES    IN    READING    FIGURES. 

1.  In  how  many  ways  may  the  figures  658  be  read? 
1st.  The  common  way,  six  hundred  and  fifty-eight. 
2d.  We  may  read,  six  hundreds,  five  tens,  and  eight  units. 
3d.   We  may  read,  sixty-five  tens  and  eight  units. 

*  Note. — This  table  is  formed  according  to  the  French  method  of  nu- 
meration.    The  English  method  gives  six  places  to  thousands,  &c. 

Quest. — Are  the  words  at  the  head  of  the  table  applicable  to  all  num* 
bers?  May  the  table  be  continued?  After  what  method  is  the  table 
formed  ?  What  is  the  difference  between  it  and  the  old  English  method  1 
12.  What  is  notation?  What  is  numeration?  In  how  many  ways  may 
the  figures  658  be  read  ? 


18  NOTATION     A.\^D     NUMERATION. 

2.  How  may  the  figures  8046  be  read  ? 

1st.  Eiglit  thousand  and  forty-six.  2d.  Eight  thousand, 
no  hundreds,  four  tens,  and  six  units.  3d.  Eighty  hundreds 
and  forty-six,  or  eighty  hundreds,  four  tens,  and  six  units. 
4th.  Eight  hundred  and  four  tens,  and  six  units. 

3.  Give  all  the  readings  of  the  number  49704. 

•  4.  Give  all  the  readings  of  the  number  740692. 

5.  Give  all  the  readings  of  the  number  99800416. 

6.  Give  all  the  readings  of  the  number  80741047. 
Note. — The  pupil  should  be  much  exercised  in  these  readings. 

He  should  remark  that  the  lowest  order  of  units  used  in  any  reading, 
whether  it  be  units,  tens,  hundreds,  &c.,  &c.,  gives  the  name  or 
denomination  to  the  part  or  whole  of  the  number  used  in  the  reading. 
We  are  now  able  to  express  any  number  whatever  in  the 
language  of  figures. 

EXAMPLES. 

1.  Write,  in  figures,  six  units  of  the  first  order.     Ans.  6. 

2.  Write,  in  figures,  eight  units  of  the  second  order. 

Ans.  SO. 

3.  Write,  in  figures,  nine  units  of  the  third  order. 

Ans.  900. 

4.  Write,  in  figures,  seven  units  of  the  fifth  order. 

Ans.  70000. 

5.  Write,  in  figures,  nine  units  of  the  first  order,  three  of 
the  third,  and  none  of  the  second.  Ans.  309. 

6.  Write,  in  figures,  eight  units  of  the  eighth  order,  six  of 
the  fifth,  seven  of  the  seventh,  five  of  the  sixth,  none  of  the 
fourth,  none  of  the  third,  one  of  the  second,  and  one  of  the 
first,  and  read  the  number.  A7is.  87560011. 

7.  Write,  in  figures,  six  quintillions,  four  hundred  and 
fifty-one  billions,   sixty-five   millions,  forty-seven   ten  thou- 

.  sands,  and  one  hundred  and  four. 

8.  Write,  in  figures,  nine  hundred  and  ninety-nine  octil- 
lions, sixty-five  millions,  eight  hundred  and  forty-one  billions, 
four  trillions,  and  eleven  nonillions. 

Quest.— How  may  the  figures  8046  be  read?  Also,  49704?  740692? 
What  gives  the  name  or  denomination  to  the  number? 


OF'  THE    DEI^OMINATION    OF  lOJMBEKS.  lO 

9.  Write,  in  figures,  sixty-five  decillions,  eight  hundred 
quadrillions,  seven  hundred  and  fifty  billions,  seven  hundred 
and  fifty-one  trillions,  nine  hundred  and  seventy-five  thousand, 
three  hundred  and  ten. 


OF  THE  DENOMINATION  OF  NUMBERS. 

13.  A  SIMPLE  NUMBER  is  ouo  which  expresses  a  collection 
of  units  of  the  same  kind,  without  expressing  the  particular 
value  of  the  unit. .   Thus,  6  and  25  are  simple  numbers. 

14.  A  DENOMINATE  NUMBER  cxprcsscs  the  kind  of  unit 
which  is  considered.  For  example,  6  dollars  is  a  denom- 
inate number,  the  unit  1  dollar  being  denominated  or  named. 

15.  When  two  numbers  have  the  same  unit,  they  are  said 
to  be  of  the  same  denomination :  and  when  two  numbers 
have  different  units,  they  are  said  to  be  of  different  denomi- 
nations. For  example,  10  dollars  and  12  dollars  are  of  the 
same  denomination ;  but  8  dollars  and  20  cents  express  num- 
bers of  different  denominations,  the  unit  of  8  dollars  being  1 
dollar,  and  of  20  cents,  1  cent.  The  kind  of  unit  always 
indicates  the  denomiiiation. 

In  simple  numbers,  the  unit  in  the  place  of  units  is  differ- 
ent from  the  unit  of  the  second  order  in  the  place  of  tens, 
and  this  last  is  different  from  that  of  the  third  order  in  the 
place  of  hundreds,  and  so  on  for  places  still  to  the  left. 
These  units,  as  we  have  seen,  have  different  names  or  de- 
nominations, viz.,  simple  units,  or  units  of  the  first  order ; 
tens,  or  units  of  the  second  order ;  hundreds,  or  units  of  the 
third  order,  &c.,  and  considered  in  this  relation  to  each  other, 
may  be  regarded  as  denominate  numbers. 

The  following  tables  show  the  various  kinds  of  denominate 

Quest. — 13.  What  is  a  simple  mimbcr?  14.  What  is  a  denominate 
number?  15.  Wlien  are  two  numbere  said  to  be  of  the  same  denomina- 
tion 1  When  of  difierent  denominations  ?  What  indicates  the  denomina- 
tion ?  In  simple  numbers,  how  are  the  units  of  the  different  places  ?  How 
do  they  compare  hi  value  with  each  other? 


20  OF    THE    DENOMINATION    OF    NUMBERS. 

numbers  in  general  use,  and  also  the  relative  values  of  tlieir 
different  units. 

OF    FEDERAL    MONEY. 

16.  Federal  money  is  the  currency  of  the  United  States. 
Its  denominations,  or  names,  are  Eagles,  Dollars,  Dimes, 
Cents,  and  Mills. 

The  coins  of  the  United  States  are  of  gold,  silver,  and  cop- 
per, and  are  of  the  following  denominations. 

1.  Gold:   eagle,  half-eagle,  quarter-eagle,  dollar. 

2.  Silver:  dollar,  half-dollar,  quarter-dollar,  dime,  half- 
dime. 

•  3.  Copper:  cent,  half-cent. 

If  a  given  quantity  of  gold  or  silver  be  divided  into  24 
equal  parts,  each  part  is  called  a  carat.  If  any  number  of 
carats  be  mixed  with  so  many  equal  carats  of  a  less  valua- 
ble metal,  that  there  be  24  carats  in  the  mixture,  then  the 
compound  is  said  to  be  as  many  carats  fine  as  it  contains 
carats  of  the  more  precious  metal,  and  to  contain  as  much 
alloy  as  it  contains  carats  of  the  baser. 

For  example,  if  20  carats  of  gold  be  mixed  with  4  of  sil- 
ver, the  mixture  is  called  gold  of  20  carats  fine,  and  4  parts 
alloy. 

17.  The  standard,  or  degree  of  purity,  of  the  gold  coin,  is 
fixed  by  Congress.  Nine  hundred  equal  parts  of  pure  gold, 
are  mixed  with  100  parts  of  alloy,  of  copper  and  silver,  (of 
which  not  more  than  one  half  must  be  silver,)  thus  forming 
1000  parts,  equal  to  each  other  in  weight.  The  silver  coins 
contain  900  parts  of  pure  silver,  and  100  parts  of  puTe 
copper.     The  copper  coins  are  of  pure  copper. 

The  eagle  contains  258  grains  of  standard  gold ;  the  dollar 
4 1 2h  grains  of  standard  silver  ;  the  cent  168  grains  of  copper. 

Quest. — 16  What  is  Federal  Money?  What  are  its  denominations? 
Of  what  are  the  coins  of  the  United  States  made  ?  What  are  their  de. 
nominations?  What  is  a  carat?  What  do  you  understand  by 'carats 
fine  ? '  What  would  be  20  carats  fine  of  gold  ?  1 7.  What  is  the  standard 
of  gold  coin  in  the  United  States?  What  of  silver?  What  of  copper? 
What  is  the  weight  of  the  eagle  ?    What  of  the  dollar  ?    What  of  the  cent  2 


OF  THE  DENOMINATION  OP  NUMBERS. 


21 


Mills. 

Cents. 

Dimes. 

Dollars. 

Eagle. 

m. 

cts. 

d. 

$ 

E. 

10 

=    1 

100 

10 

=    1 

1000 

100 

10 

=    1 

10000 

1000 

100 

10 

=  1 

This  table  is  read,  ten  mills  make  one  cent,  ten  cents  one 
dime,  ten  dimes  one  dollar,  ten  dollars  one  eagle.  In  this 
table,  ten  units  of  each  denomination  make  one  unit  of  the 
denomination  next  higher,  the  same  as  in  simple  numbers. 

In  expressing  Federal  Money  in  the  language  of  figures, 
the  dollars  are  separated  from  the  cents  and  mills  by  a 
comma :  thus,  36,645  is  read,  36  dollars,  64  cents,  5  mills ; 
but  may  also  be  read,  36  dollars,  6  dimes,  4  cents,  5  mills ; 
375,043  is  read  375  dollars,  4  cents,  3  mills. 


ENGLISH    CURRENCY. 

18.  The  relative  proportion  between  gold  and  silver  in  the 
English  coins,  according  to  the  mint  regulations,  both  for  the 
old  and  new  coinage,  is  as  follows :  in  the  old  coinage,  a 
pound  of  gold  is  worth  15.2096  times  a  pound  of  silver.  In 
the  new  coinage,  a  pound  of  gold  is  worth  14.2878  times  a 
pound  of  silver. 

A  standard  gold  coin  is  composed  of  22  parts  of  pure  gold 
and  2  parts  of  copper. 

A  standard  silver  coin  is  composed  of  224  parts  of  pure 
silver  and  18  parts  of  copper. 

In  the  copper  coin  24  pence  make  one  pound  avoirdupois. 

-m • 

Quest. — Repeat  the  table  of  Federal  money.  How  many  units  of  eacli 
denomination  make  one  of  the  next  higher  ?  In  expressing  Federal  money 
in  figures,  how  are  the  dollars  separated  from  the  cents  ?  What  place  do 
the  mills  occupy,  counting  from  the  comma?  18.  What  is  the  relative  pro- 
portion between  gold  and  silver  in  the  old  and  new  coinage  of  Ejiglish 
money?  What  is  the  standard  of  the  English  gold?  Of  the  silver?  What 
is  the  weight  of  the  English  penny  ? 


22 


OF    THE    DENOMINATION    OF    NUMBERS. 


i 

1 

1 
1 

i 

a 

.s 

i 

{ 
1 

.5^ 

la 

if 

1 
1. 

1 

far. 

d. 

8. 

£ 

2 

—  1 

4 

2 

=zl 

16 

8 

4 

=  1 

24 

12 

6 

U 

=  1 

48 

24 

12 

3 

2 

=  1 

120 

60 

30 

7^ 

5 

2h 

=  1 

240 

120 

60 

15 

10 

5 

2 

=1 

336 

168 

84 

21 

14 

7 

24 

If 

=1 

480 

S40 

120 

30 

20 

10 

4 

2 

1^ 

—I 

504 

252 

126 

3U 

21 

101 

4^ 

2tV 

2f 

1* 

=1 

960 

480 

240 

60 

40 

20 

8 

4 

2 

1^ 

—  I 

1008 

504 

252 

63 

42 

21 

8f 

4^ 

3 

2tV 

2 

l^TT 

=1 

AVOIRDUPOIS    WEIGHT. 


19.  The  standard  avoirdupois  pound  of  the  United  States, 
as  determined  by  Mr.  Hassler,  is  the  weight  of  27.7015  cubic 
inches  of  distilled  water  weighed  in  air. 

By  this  weight  are  weighed  all  coarse  articles,  such  as 
hay,  grain,  chandlers'  wares,  and  all  the  metals,  excepting 
gold  and  silver. 

In  this  weight  the  words  gross  and  net  are  used.  Gross 
is  the  weight  of  the  goods,  with  the  boxes,  casks,  or  bags  in 
which  they  are  contained.  Net  is  the  weight  of  the  goods 
only ;  or  what  remains  after  deducting  from  the  gross  weight 
the  weight  of  the  boxes,  casks,  or  bags. 

An  hundred  weight,  in  its  general  sense,  means  112  pounds, 
as  appears  in  the  table.  But  by  the  laws  of  the  United  States 
it  is  fixed  at  100  pounds,  (See  Preface.) 


Quest. — Repeat  the  table  of  English  money.  19.  What  is  the  standard 
avoirdupois  pound  of  the  United  States  ?  For  what  is  this  weight  used  ? 
What  is  the  meaning  of  the  terms  gi'oss  and  net  ?  What  is  a  hundred 
weight  ?     How  are  goods  now  generally  bought  and  sold  ? 


OF   THE    DENOMINATION    OF    NUMBERS. 


23 


TABLE. 


Drams. 

Ounces. 

Pounds. 

(Quarters. 

Hundreds. 

Tons. 

dr. 

oz. 

lb. 

gr. 

cwt. 

T. 

16 

256 

7168 

28672 

573440 

—    1 

16 

448 

1792 

35840 

=    1 

28 
-    112 
2240 

—    1 

4 

80 

=z    1 

20 

=  1 

TROY    WEIGHT. 


20.  By  this  weight  are  weighed  gold,  silver,  jewels,  and 
some  liquids. 

The  standard  troy  pound  of  the  United  States,  as  deter- 
mined by  Mr.  Hassler,  is  the  weight  of  22.794377  cubic 
inches  of  distilled  water  weighed  in  air.  Hence,  the  poimd 
is  less  than  the  pound  avoirdupois. 


Grains. 

Penn^eights. 

Ounces. 

Pounds. 

gr. 

pwt. 

oz. 

lb. 

24 

480 
5760 

=.    1 

20 

240 

=    1 
12      • 

=  1 

COMPARISON    WITH    AVOIRDUPOIS    WEIGHT. 

7000    troy  grains    =       1  lb.  avoirdupois. 
175    troy  pounds  r^  144  lbs.        " 
175    troy  ounces  =  192  oz.         " 
4371  troy  grains    =:      1  oz.         " 


Quest. — Repeat  the  table  of  avoirdupois  weight.  20.  What  articles 
are  weighed  by  troy  weight  ?  What  is  the  standard  troy  pound  of  the 
United  States  ?  Is  it  greater  or  less  than  the  avoirdupois  pound  ?  Repeat 
the  table  of  troy  weight.    How  does  it  compare  with  avoirdupois  weight  ? 


24 


OF    THE    DENOMINATION    OF    NUMBERS. 


APOTHECARIES  WEIGHT. 

21.  This  weight  is  used  by  apothecaries  and  druggists  in 
mixing  their  medicines.  They,  however,  buy  and  sell  their 
drugs  by  avoirdupois  weight.  The  pound  and  ounce  are  the 
same  as  the  pound  and  ounce  in  troy  weight.  The  difference 
between  the  two  weights  consists  in  the  different  divisions 
and  subdivisions  of  the  ounce. 


Grains. 

Scruples. 

Drams. 

Ounces. 

Pound. 

gr. 

9 

3 

5 

flj 

20 

60 

480 

6760 

=  1 
3 

24 

288 

=  1 

8 
96 

=  1 

12 

=  1 

FOREIGN    WEIGHTS. 

22.  The  foreign  weights  differ  sGimewhat  from  ours. 

i  pound  avoirdupois,  English  =  27.7274  cubic  inches  dis- 
tilled water. 

1  pound  troy,  English  =  22.815689  cubic  inches  distilled 
water. 


OLD    FRENCH    SYSTEM. 


1  livre   =16  onces 
1  once  =    8  gros 
1  gros    =:  72  grains 
1  grain 


=    1.0780  lb.  avoirdupois. 
=    1.0780  oz.         " 
r=:  58.9548  grains  troy. 
=    0.8188         " 


Quest. — 21.  By  whom  is  apothecaries  weight  used?  By  what  weight 
do  druggists  buy  and  sell  their  drugs?  In  what  respects  is  the  weight 
similar  to  troy?  In  what  is  the  difference?  Repeat  the  table  of  apothe- 
caries weight.  22.  What  is  the  value  of  the  English  pound  avoirdupois? 
Of  the  English  pound  troy  ?     What  is  the  old  French  system  of  weights  1 


OF  THE  DENOMINATION  OF  NUMBERS. 


2D 


NEW    FRENCH    SYSTEM. 

23,  The  basis  of  this  system  of  weights  is  the  weight  in 
vacuo  of  a  cubic  decimetre  of  distilled  water.  This  weight 
is  called  a  kilogramme,  and  is  the  unit  of  the  French  system. 
It  is  equal  to  2.204737  pounds  avoirdupois.  (For  the  value 
of  a  decimetre,  see  table  of  linear  measure,  French,  page  29.) 
The  one-thousandth  part  of  a  kilogramme  is  called  a  gramme, 
and  the  one-thousandth  part  of  a  gramme  is  called  a  milli- 
gramme. 

The  divisions  are  made  on  the  decimal  principle,  and 
are  of  the  following  denominations  : 


Milligramme. 

Centi- 
gramme. 

Deci- 
gramme. 

Gramme. 

Deca- 
gramme. 

Hecto- 
gram'e. 

Kilo- 
gr*me. 

d    • 

Mil- 
lier. 

10 

=  1 

^ 

100 

10 

=  1 

a 

1 

1000 

100 

10 

=   1 

10000 

1000 

100 

10 

=    1 

s 

100000 

10000 

1000 

100 

10 

=    1 

^ 

1000000 

100000 

10000 

1000 

100 

10 

=  1 

10000000 

1000000 

100000 

10000 

1000 

100 

10 

» 

100000000 

10000000 

1000000 

100000 

10000 

1000 

100 

=1 

3 

1000000000 

100000000 

10000000 

1000000 

100000,10000 

1000 

10 

=  1 

COMPARISON    OF    WEIGHTS. 

English,  1  pound  =  1.000936  pounds  avoirdupois. 

French,  1  kilogramme  =  2.204737      "  " 

Spanish,  1  pound  =  1.0152           "  « 

Swedish,  1  pound  =  0.9376           "  « 

Austrian,  1  pound  =  1.2351           "  " 

Prussian,  1  pound  —  1.0333           "  " 


Quest. — 23.  What  is  the  basis  of  the  new  French  system  ?  Repeat 
the  table.  How  do  the  weights  of  different  ooimtries  compare  with 
ours? 

3 


26 


OF    THE    DENOMINATION    OF    NUMBERS. 


ENGLISH    WOOL    WEIGHT. 


24.  The  following  is  the  table  of  wool  weight  in  England. 
As  yet,  many  of  the  denominations  have  not  been  much  used 
in  this  country ;  but  as  we  are  now  exporting  wool  to  Eng- 
land, they  must  soon  be  generally  introduced. 


Pounds. 

Cloves. 

Stones. 

Tods. 

Weys. 

Sacks. 

Last. 

7 

=   1 

14 

2 

=    1 

28 

4 

2 

=    1 

182 

26 

13 

64 

=    1 

364 

52 

26 

13 

2 

=    1 

4368 

624 

312 

156 

24 

12 

=  1 

CLOTH    MEASURE. 


25.  Cloth  measure  is  used  by  woollen  and  linen  drapers. 
Hollands  are  measured  in  English  ells,  and  tapestry  by  the 
French  ell ;  woollens,  linens,  silks,  tape,  &c.,  by  the  yard. 


TABLE. 


Inches. 

Nails. 

Quarters. 

Ells  Flemish. 

Yards. 

Ells  English. 

Ell  French. 

in. 

na. 

qr. 

E.Fl. 

yd. 

E.E. 

E.  Fr. 

2i 

=    1 

9 

4 

-1 

27 

12 

3 

=  1 

36 

16 

4 

H 

=  1 

45 

20 

5 

If 

n 

=  1 

64 

24 

6 

2 

H 

1^ 

=  ^ 

Quest. — ^24.  Is  English  wool  weight  yet  in  use  in  this  country?  Repeat 
the  table  of  wool  weight.  25.  By  whom  is  cloth  measure  used  ?  How  are 
hoUands  measured  ?  Tapestry  ?  Repeat  the  table. 


err    THE    DENOMrNATION    OF    NUMBERS. 


27 


LONG    MEASURE. 

26.  This  measure  is  used  to  measure  distances,  lengths, 
breadths,  heights,  depths,  &c.  Gunter's  chain  is  generally 
used  by  surveyors  in  measuring  land.  A  standard  measure 
has  been  adopted  by  the  United  States,  copies  of  which  are 
distributed  in  various  parts  of  the  country.  This  standard  is 
a  brass  rod,  one  yard  or  3  feet  long. 

TABLE. 


Inches. 

Gunter's 
Link. 

Feet. 

Yards. 

Fathoms. 

Rods. 

Gnnter's 
Chain. 

Fur- 
longs. 

Mile. 

in. 

I. 

ft' 

yd. 

^d. 

c. 

fur. 

mi. 

12 

36 

72 

198 

792 

7920 

63360 

=  1 

m 

25 

100 
1000 
8000 

=  I 
3 
6 

16i 
66 
660 
5280 

=  1 
2 

22 

220 

1760 

1? 

110 

880 

=  1 

4 

40 

320 

=  1 

'    10 

80 

=  1 

8 

=  1 

FOREIGN  MEASURES  OF  LENGTH. 

27.  The  imperial  standard  yard  of  Great  Britain  is  the  one 
from  which  our  yard  is  taken.  It  is  referred  to  a  natural 
Standard,  viz.,  to  the  distance  between  the  axis  of  suspension 
and  the  centre  of  oscillation  of  a  pendulum  which  shall  vi- 
brate seconds  in  vacuo,  in  London,  at  the  level  of  the  sea. 
This  distance  is  declared  to  be  39.1393  imperialinches ;  that 
is,  1  imperial  yard  and  3.1393  inches. 


Quest. — 26.  For  what  is  long  measure  used?  For  what  is  Gunter's 
chain  used  ?  Repeat  the  table  of  long  measure.  27.  What  is  the  stan- 
dard of  the  English  imperial  yard  ?    What  is  its  length  ? 


28 


OF    THE    DENOMINATION    OF    NUMBERS. 


OLD    FRENCH    SYSTEM. 


1  point 

1  line    =  1*^  points 
1  inch  ==  IS  lines 
1  foot    =12  inches 


=    0.0074  U.  S.  inches. 
=    0.08884 
=     1.06604 
=  12.7925 


=  1.298  yd 
=  2.132  " 


4861 


1  ell      =  43  tn.  10  lines  =  46.728 

1  toise  =    6  feet  =  76.755 

1  perch  (Parl?s)  =  18  feet. 

1  perch  (royal)  =  22  feet. 

1  league  (common)  25  to  a  degree  =  2280  toises 

yards  =  2.76  miles. 
1  league  (post)  =  2000  toises  =  4264  yards  =  2.42  miles. 
1  fathom  (Brasse)  =  5  feet  French  =  63.963  inches,  or  5\ 

feet  English,  nearly. 
1  cable  length  =100  toises  =  120  fathoms  French  =  106| 

fathoms  English. 

TABLE     FOR     REDUCING    OLD    FRENCH    MEASURES    TO    UNITED 

STATES  'measures. 

(According  to  Mr.  Hassler's  comparison.) 


French 

U.  States 

French  ft. 

U.  S.  feet 

French 

U.  States 

French 

U.  states 

feet. 

inches. 

or  inches. 

or  inches. 

lines. 

inches. 

points. 

inches. 

1 

12.7925 

1 

1.0660 

1 

0.0888 

1 

0.0074 

2 

25.5850 

2 

2.1321 

2 

0.1777 

2 

0.0148 

3 

38.3775 

3 

3.1981 

3 

0.2665 

3 

t).0222 

4 

51.1700 

4 

4.2642 

4 

0.3554 

4 

0.0296 

5 

63.9625 

5 

5.3302 

5 

0.4442 

.  5 

0.0370 

6 

76.7550 

6 

6.3963, 

6 

0.5330 

6 

0.0444 

7 

89.5475 

7 

7.4623 

7 

0.6219 

7 

0.0518 

8 

102.3400 

8 

8.5283 

8 

0.7107 

8 

0.0592 

9 

115.1325| 

9 

9.5944 

9 

0.7995 

9 

0.0666 

10 

127.9250; 

10 

10.66041 

10 

0.8884 

10 

0.0740 

11 

140.7175' 

1 

11 

11.7265 

11 

0.9772 

11 

0.0814 

NEW    FRENCH    SYSTEM. 

28.  The  basis  of  the  new  French  system  of  measures  is  the 

Quest. — What  is  the  old  French  long  measure  ?    28.  What  is  the  basia 
of  the  new  French  system  ? 


OF  THE  DENOMINATION  OF  NUMBERS. 


29 


measure  of  the  meridian  of  the  earth,  a  quadrant  of  which  is 
10,000,000  metres,  measured  at  the  temperature  of  32*^  Fahr. 
The  muhiples  and  divisions  of  it  are  decimals,  viz. :  1  me- 
tre =  10  decimetres  =:  100  centimetres  =  1000  millimetres 
=  39.3809171  United  States  inches,  or  3.28174  feet. 

Road  measure.  Myriametre  =  10,000  metres.  Kilometre 
=  1000  metres.  Decametre  =  10  metres.  Metre  =  0.51317 
toise. 


TABLE    FOR    REDUCING    METRES    TO    INCHES. 
(According  to  Mr.  Hassler's  comparisons ;  1  metre  =  39.3809171  inches.) 


Metres, 

Inches. 

Metres. 

Inches. 

Metres. 

Inches. 

Metres. 

Inches. 

0.001 

0.039381 

0.026 

1.023904 

0.051 

2.008427 

0.076 

2.992950 

2 

0.078762 

:  27 

1.063285 

*   52 

^S*.  04.7808 

77 

3.032331 

3 

0.118143 

28 

1.102666 

53 

2.087189 

78 

3.071712 

4 

0.157524 

29 

1.142047 

54 

2.126570 

79 

3.111093 

6 

0.196905 

0.030 

1.181428 

55 

2.165950 

0.080 

3.150474 

6 

0.236286 

31 

1.220809 

.  56 

2.205331 

81 

3.189855 

7 

0.275666 

32 

1.260189 

57 

2.244712 

82 

3.229236 

8 

0.315047 

33 

1.299570 

58 

2.284093 

83 

3.268617 

9 

0.354428 

34 

1.338951 

59 

2.323474 

84 

3.307998 

0.010 

0.393809 

35 

1.378332 

0.060 

2.362855 

85 

3.347379 

11 

0.433190 

36 

1.417713 

61 

2.402236 

86 

3.386759 

12 

0.472571 

37 

1.457094 

62 

2.441617 

87 

3.426140 

13 

0.511952 

38 

1.496475 

63 

2.480998 

88 

3.465521 

14 

0.551333 

39 

1.535856 

64 

2.520379 

89 

3.504902 

15 

0.590714 
0.630095 

0.040 

1.575237 

65 

2.559760 

0.090  3.544282 

16 

41 

1.614618 

66 

2.599141 

91  3.583663 

17 

0.669476 

42 

1.653999 

67 

2.638522 

92  3.623044 

18 

0.708855 

43 

1.693379 

68 

2.677903 

93,3.662425 

19 

0.748237 

44 

1.732760 

69 

2.717283 

9413.701806 

0.020 

0.787618 

45 

1.772141 

e.070 

2.756664 

95  3.741187 

21 

0.826999 

46 

1.811522 

71 

2.796045 

96  3.780568 

22 

0.866380 

47 

1.850903 

72  2.835426 

97|  3.819949 

23 

0.905761 

48 

1.890284 

73  2.874807 

98  3.859330 

24 

0.945142 

49 

1.929665 

74  2.914188 

99,3.898711 

25  0.984593 

0.050 

1.969046 

75  2.953569 

0.100,3.938092 

Quest. — ^What  are  th^  multiples  and  divisions  of  it?     Repeat  the  table 
of  road  measure 


30       OP  THE  DENOMINATION  OF  NUMBEES. 

Austrian,       1  foot  =  12.448  U.  S.  inclies  =  1.03737  foot. 
Prussian,    )      ^ 

RWa»U'^°°'  =  '^-^^'  "  "  ='-^'°^  " 
Swedish,        1  foot  =  11.690      «         «        =  0.974145  " 

r  1  foot  =  11.034  "  "  =  0.9195  " 
Spanish,     <  league  (royal)  =  25000  Span.  ft.  =z  41  miles  >  > 

(     "   (common)  =  19800       "        =3^     "     il 

SQUARE    MEASURE. 

29.  Square  measure  is  used  for  measuring  all  kinds  of 
superficies,  such  as  land,  paving,  flooring,  plastering,  and 
every  thing  which  has  length  and  breadth. 


TABLE. 

riif0      /yc-^             SO 

Square  inches. 

Square 
links. 

Square 
feet. 

Square 
yards. 

Poles  or 
perches. 

2  2 

8 

< 

Sg.  in. 

Sq.l. 

Sq.ft. 

'Sg.  yd. 

P. 

Sg.e. 

R. 

A. 

M. 

62MI 

=  1 

144 

O  322 

=  1 

1296 

20^ 

9 

=  1 

39204 

625 

2721 

30| 

=  1 

627264 

10000 

4356 

484 

16 

=  1 

1568160 

25000 

10890 

1210 

40 

21 

=  1 

6272640 

100000 

43560 

4840 

160 

10 

4 

—  \ 

4014489600 

64000000 

27878400  3097600 

102400 

6400!l560'640 

—  \ 

1 

'     ^/ra 

FRENCH    SUPERFICIAL    MEASURE,    OLD    SYSTEM. 

1  square  inch  =  1.1364  U.  S.  square  inches. 

1  arpent  (Paris)  =  100  square  perches,  (Paris,)  or  900  square 

toises  =  4088  square  yards,  or  |ths  of  an  acre,  nearly. 
1  arpent  (woodland)  =  100  square  perches  (royal)  =  6108 

square  yards,  or  1  acre,  1  rood,  1  perch. 


Quest. — 29.- For  what  is  square  measure  used?     Repeat  the  table. 
What  is  the  old  French  system  of  square  measure  ? 


OF  THE  DENOMINATION  OF  NUMBERS. 


31 


NEW    SYSTEM. 

1  are    =  100  square  metres  =  119.665  square  yards. 
10  ares  =  1  decare.     10  decares  =  1  hecatare. 

CUBIC    OR    SOLID    MEASURE. 

30.  Forty  cubic  feet  of  round  timber,  or  50  solid  feet  of 
square  timber,  make  1  ton.  A  cord  of  wood  is  a  pile  4  feet 
high,  4  feet  wide,  and  8  feet  long,  and  consequently  contains 
128  solid  feet.  A  cord  foot  is  one  foot  in  length  of  the  pile 
which  makes  a  cord.     It  contains  16  solid  feet. 


Cubic  inches. 

Cubic  feet. 

Cubic  yards. 

Cubic  rods. 

Cubic 
furlongs. 

Cub. 
mile. 

S.m. 

S.ft. 

S.yd. 

S.rd, 

S.fur. 

S.mi. 

1728 

46656 

7762392 

496793088000 

254358061056000 

=  1 

27 

4492  J 

287496000 

147197952000 

=  1 

1661 

10648000 

5451776000 

=  1 

64000 
32768000 

=  1 
612 

=  1 

FRENCH    SOLID    MEASURE. 

.  cubic  foot  =  2093.470  cubic  inches  of  the  U.  States. 

1  cubic  decimetre  =  61.074664  "  "  " 

1  stere  =  1  cubic  metre  =  61074.664  cubic  inches  =  35.375 
cubic  feet  =  1.309  cubic  yards. 


LIQUID    MEASURE    OF    THE    UNITED    STATES. 

31.  The  standard  gallon  of  the  United  States  is  the  wine 
gallon,  which  measures  231  cubic  inches,  and  contains,  as 

Quest. — What  the  new  system  ?  30.  How  much  timber  makes  a  ton  ! 
What  is  a  cord  of  wood  ?  How  many  solid  feet  does  it  contain  ?  What 
is  a  cord  foot  ?  How  many  solid  feet  does  it  contain  ?  Repeat  the  table  of 
cubic  or  solid  measure.  Repeat  the  table  of  French  solid  measure.  31 
What  is  the  standard  gallon  of  the  United  States  ? 


82 


OF  THE  DENOMINATION  OF  NUMBERS. 


determined  by  Mr.  Hassler,  8.3388822  pounds  avoirdupois 
of  distilled  water. 


Cubic  inches. 

Gills. 

Pints. 

auarts. 

Gallon. 

S.  in. 

ffi- 

pt. 

gt. 

gal. 

7^ 
28J"' 
57|  ' 
231 

=  1 

4 

8 

32 

=  1 
2 

8 

4 

=  1 

DRY    MEASURE. 


32.  The  standard  bushel  of  the  United  States  is  the  Wm 
cAe^^er  bushel,  which  measures  2150.4  cubic  inches,  and  con 
tains  77.627413  pounds  avoirdupois  of  distilled  water. 


Cubic  inches. 

Pints. 

Quarts. 

Gallons. 

Pecks. 

Bushels. 

S.  in. 

pt. 

gt. 

gal. 

pk. 

bu. 

33| 

67i 

268j 

5371 

2150f 

=  1 

2 

8 

16 

64 

=  1 

4 

8 

32 

=  1 

2 

8 

=  1 

4 

=  1 

.     FOREIGN    MEASURES. 

33.  The  British  imperial  gallon  contains  10  pounds  avoirdu- 
pois of  distilled  water  weighed  in  air,  and  measures  277.274 
cubic  inches.  The  same  measure  is  now  used  for  liquids  as 
for  dry  articles  which  are  not  measured  by  heaped  measure. 

Quest. — Repeat  the  table  of  liquid  measure.  32.  What  is  the  standard 
bushel  of  the  United  States  ?  How  many  cubic  inches  does  it  contain  ? 
Repeat  the  table  of  dry  measure.  33.  What  is  the  standard  of  the  British 
hnperial  gallon? 


OF  THE  DENOMINATION  OF  NUMBERS. 


33 


For  the  latter,  the  bushel  is  heaped  in  the  form  of  a  cone,  not 
less  than  6  inches  high,  the  base  being  9^  inches. 

French,  1  litre  =:  1  cubic  decimetre  =  61.074  U.  S.  cubic 
inches  =  1.057  U.  S.  quarts,  wine  measure  = 
1.761  imperial  pints  of  Great  Britain. 

1  boisseau  =:13  litres  =:  793.364  cubic  inches  = 
3.4349  gallons. 

1  pinte  z=z  0.931  litre  =  56. 8L7  cubic  inches  = 
0.98397  quarts. 

Spanish,   1  wine  arroba  =  4.2455  gallons. 

1  fanega  (corn  measure)  =  1.593  bushels. 

ENGLISH  ALE  AND  BEER  MEASURE. 

34.  The  following  is  the  English  beer  measure..   By  it 
all  malt  liquors  and  water  are  measured. 

TABLE. 


Cubic  inches. 

a 

i 

o 

U 
g 

i 
1 

s 
IS 

2 

2 
1 

1 

1 

i 

d 

S 

S.  in. 

pt. 

qt. 

gal. 

^r. 

K. 

bar. 

hkd. 

pun. 

B. 

tun. 

34. 659 J 
69.3181 
277.274^ 
2495.466 
4990.932 
9981.864 
14972.796 
19963.728 
29945.592 
59891,184 

=  1 

2 

8 

72 

144 

288 
432 
576 
864 
1728 

=  I 

4 
36 

72 
144 
216 
288 
432 
864 

=  1 

9 

18 
36 
54 

72 
108 
216 

=  1 

2 
4 
6 
8 
12 
24 

=  1 
2 
3 
4 
6 
12 

=  1 

2 
3 
6 

=1 

n 

2 
4 

=  1 

3 

=  1 
2 

=  1 

Quest. — ^What  is  heaped  measure  ? ,  What  is  the   French  measure  ? 
34.  What  is  measured  by  English  beer  measure  ?     Repeat  the  table 

2* 


34 


OF    THE    DENOMINATION    OF    NUMBERS. 


ENGLISH    WINE    MEASURE. 


35.  The  following  is  the  English  wine  measure.  All  the 
denominations  of  it  are  not  generally  used  in  this  country. 
By  this  measure  all  wines,  brandy,  rum,  and  distilled  liquors 
ar^  bought  and  sold. 


Cubic  inches. 

3 

1 

j 

1 

< 

a 

1 

p; 

^ 

w 

fl 

5 

S.  in. 

si- 

pt. 

qt. 

gal. 

ank. 

run. 

bar. 

tier. 

hhd. 

/;m?i. 

;,». 

twn. 

8.664f| 

=  1 

a4.659i 

4 

■=z  1 

69.318i 

8 

2=1 

277.274 

32 

8,       4 

=  1 

2772.740 

320 

80,     40 

10 

=  1 

4990.932 

576 

144 1     72 

18 

u 

—  I 

8734.131 

1008 

252    126;3U 

q  3 

1? 

—  I 

11645.508 

1344|  336    168 

42 

4^ 

^ 

14 

=  1 

17468.262 

2016:  504    252 

63 

6^ 

4 

2 

u 

=  1 

23291.016 

2688|  672    336 

84 

8| 

4f 

2f 

2 

Hl^i 

34936.524 

40321008    504 

126 

12^ 

7 

4 

3 

2 

u 

::=:1 

69873.048 

8064  2016  1008|252 

25^ 

14 

8 

6 

4 

3 

2 

=.1 

ENGLISH  CORN  OR  DRY  MEASURE. 

36.  Dry  measure  is  used  for  all  dry  commodities,  such  as 
wheat,  barley,  beans,  coal,  oysters,  &c.  The  following  is 
the  English  table,  all  the  denominations  of  which  are  not  in 
general  use  in  this  country.  The  standard  bushel  is  a  cyl- 
inder 18.789  inches  in  the  interior  diameter,  and  8  inches  in 
depth,  and  consequently  contains  2218.192  cubic  inches. 


Quest. — 35.  What  liquids  are  measured  by  wine  measure  ?  Repeat  the 
table.  36.  What  articles  are  measiu-ed  by  corn  measure  ?  What  is  the 
standard  busliel  ?     How  many  cubic  inches  does  it  contain  ? 


OF   THE    DENOMINATION    OF    NUMBERS. 
TABLE. 


35 


Cubic  inches. 

f 

09 

1 
1 

13  w 
II 

1 

1 

j 

1 

I 

S.  in. 

pt. 

qt. 

gal. 

pk. 

bu. 

qr. 

34.659i 

~  1 

69.318^ 

2 

=  1 

138.637 

4 

2 

=  1 

277.274 

8 

4 

2 

=  1 

554.548 

16 

8 

4 

2 

=:  I 

2218.192 

64 

32 

16 

8 

'4 

—I 

4436.384 

128 

64 

32 

16 

8 

2 

=  1 

8872.768 

256 

128 

64 

32 

16 

4 

2 

=  1 

17745.536 

512 

256'   128 

64 

32 

8 

4 

2 

—  \ 

88727.680 

2560 

1280;  640 

320 

160 
320 

40 

20 

10 

5 

=  1 

177455.360 

5120 

2560  1280 

640 

80 

40 

20 

10 

2 

=1 

OLD    ANB    NEW    ENGLISH    COAL    MEASURE. 

37.  By  act  of  Parliament  passed  in  1831,  all  coals  sold 
within  25  miles  of  the  Post  Office  in  London,  are  to  be  sold 
by  weight.  One  sack  weighs  2  cwt.  or  224  lbs. ;  consequently, 
10  sacks  make  1  ton.  Twelve  sacks  make  a  London  chal- 
dron of  36  bushels,  while  it  takes  79^  bushels  to  make  a 
Newcastle  chaldron,  as  shown  by  the  table. 


Pounds 

Pecks. 

Busliels. 

Sacks. 

Vats,  or 

London 

Newc. 

Keels. 

Scows. 

Iship 

weight. 

strikes. 

chaldron. 

chald. 

load. 

18| 

=  1 

74f 

4 

=  1 

224 

12 

3 

=  1 

672 

36 

9 

3 

—  \ 

2688 

144 

36 

12 

4 

=  1 

5936 

318 

791 

26-i 

8| 

17| 

=  1 

47488 

2544 

636 

212 

70f 

8 

=  1 

56448 

3024 

756 

252 

84 

21 

Q27 

iM=i 

949760 

50880 

12720   4240 

1 

4131 

353i 

160 

20     1  13|1 

=  1 

Quest. — Repeat  the  table.    37.  What  was  established  by  act  of  Parlia- 
ment \     Repeat  the  table  of  coal  measure. 


36 


OF  THE  DENOMINATION  OF  NUMBERS. 


M 

EASURI 

3    OF    TIME. 

Thirds. 

Seconds. 

Minutes. 

Hours. 

Days. 

Weeks. 

Months. 

Year. 

thirds. 

sec. 

m. 

hr. 

da. 

wk. 

mo. 

yr. 

60 

—  1 

3600 

60 

=:1 

216000 

3600 

60 

—  1 

5184000 

86400 

1440 

24 

—  I 

36288000 

604800 

10080 

168 

7 

—  1 

145152000 

2419200 

40320 

672 

28 

4 

—  I 

1893456000 

31557600 

525960 

8766 

365^ 

52A 

13rf2 

=  \ 

38.  The  whole  days  only  are  reckoned.  The  odd  six 
hours,  by  accumulating  for  4  years,  make  one  day,  so  that 
every  fourth  year  contains  366  days.  This  is  called  the 
Bissextile,  or  Leap  year.  The  leap  years  may  always  be 
known  by  this,  that  the  numbers  which  express  them  are 
exactly  divisible  by  4.  Thus,  1840,  1844,  1848,  &c.,  are 
all  leap  years. 

Although  the  year  is  reckoned  at  365^a.  6Ar.,  it  is  in  fact 
but  365Ja.  bJir.  48m.  ASsec,  and  the  difference  by  acumula- 
ting  for  100  years  makes  about  1  day,  so  that  the  centennial 
years,  though  divisible  by  4,  are  not  leap  years. 

The  year  is  also  divided  into  12  calendar  months,  which 
contain  an  unequal  number  of  days. 


JSTames. 

JVo 

of  Days. 

1  month  January 

- 

31 

2 

'       February     - 

- 

28 

3 

'       March 

- 

31 

4 

'       April  - 

- 

30 

5 

'       May    - 

- 

31 

6 

'       June   - 

- 

30 

7 

'       July    . 

'- 

31 

8 

*       August 

- 

31 

9 

*       September  - 

- 

30 

10 

'       October       - 

- 

31 

11 

*       November  - 

- 

30 

12 

*       December  - 

Total 

31 

365 

Quest. — 38.  Repeat  the  table  of  time.  What  is  the  length  of  a  yearl 
What  is  done  with  the  quarter  of  a  day  ?  How  do  you  determine  the 
leap  years?    What  years  that  are  divisible  by  four  are  not  leap  years'* 


OF  THE  DENOMINATION  OF  NUMBERS. 


37 


The  additional  day,  when  it  occurs,  is  added  to  the  month 
of  February,  so  that  this  month  has  29  days  in  the  Leap 
year. 

Thirty  days  hath  September, 

April,  June,  and  November ; 

All  the  rest  have  thirty-one. 

Excepting  February,  twenty-eight  alone. 

But  Leap  year  coming  once  in  four, 

February  then  has  one  day  more. 


TABLE,  SHOWING  THE  NUMBER  OF  DAYS  FROM  ANY  DAY  OF 
ONE  MONTH  TO  THE  SAME  DAY  OF  ANY  OTHER  MONTH  IN 
THE  SAME  YEAR. 


From  any 
day  of 

To  the  same  day. 

From  any 
day  of 

Jan. 

Feb. 

Mar. 

April 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

January 

365 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

Jan. 

February 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

Feb. 

March 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

March 

April 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

April 

May 

245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

May 

June 

214 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183 

June 

July 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

July 

August 

153 

184 

212 

243 

273 

304 

334 

365 

31 

62 

92 

122 

August 

Sept. 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

Sept. 

October 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

Oct. 

Nov. 

61 

92 

120 

151 

181 

212 

242 

273 

304 

3^4 

365 

30 

Nov. 

Dec. 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

Dec. 

39.  The  months  counted  from  any  day  of,  are  arranged  in 
the  left  hand  vertical  column ;  those  counted  to  the  same  day, 
are  in  the  upper  horizontal  line  :  the  days  between  these 
periods  are  found  in  the  angle  of  intersection,  in  the  same 
way  as  in  a  common  table  of  multiplication.  If  the^end  of 
February  be  included  between  the  two  points  of  time,  a  day 
must  be  added  in  leap  years.     Suppose,  for  example,  it  were 

Quest. — 38.  What  are  the  calendar  months?  How  many  days  does  each 
contain  ?  What  is  done  with  the  odd  day  in  leap  year  ?  Repeat  the  verse 
which  indicates  the  number  of  days  in  each  month  of  the  year.  39.  What 
is  the  object  of  this  table  ? 


38        OF  THE  DENOMINATION  OF  NUMBERS. 

required  to  know  the  number  of  days  from  the  fourth  of  March 
to  the  fifteenth  of  August.  In  the  left  hand  vertical  column 
find  March,  and  then  referring  to  the  intersection  of  a  hori- 
zontal line  drawn  from  March,  with  the  column  under  August, 
we  find  153,  which  is  the  number  of  days  from  the  fourth 
{or  any  other)  day  of  March  to  the  fourth  {or  same)  day  of 
August ;  but  as  we  want  the  time  to  the  fifteenth  of  August, 
11  days  {the  difference  between  4  and  15)  must  be  added  to 
153,  which  shows  that  164  is  the  number  of  days  between 
the  fourth  of  March  and  the  fifteenth  of  August. 

Again,  required  the  number  of  days  between  the  tenth  of 
October  and  the  third  of  June,  in  the  following  year.  Oppo- 
site to  October  and  under  June,  we  find  243,  which  is  the 
number  of  days  from  the  tenth  of  October  to  the  tenth  of  June  ; 
but  as  we  sought  the  time  to  the  third  only,  which  is  7  days 
earlier,  we  must  deduct  7  from  243,  leaving  236,  the  number 
of  days  required ;  and  so  of  others. 

DIVISION    OF    THE    CIRCLE MEASURE    OF    TIME. 


The  geographical  division  of  any  line  drawn  round  the  cir- 
cumference of  the  Earth. 

Diurnal  motion  of  the 
Earth    reduced    to 
time. 

60  seconds,    1  minute  -         -         -         -         - 
60  minutes,    1  degree  -         -         -         -         - 
15  degress,    -^  sign  of  the  zodiac  -         -         - 
30  degrees,    1  sign  of  the  zodiac  -         -         - 
90  degrees,    1  quadrant         -         -         - 
4  quadrants,  or  360  degrees,  1  great  circle  - 

=    4  seconds. 
=    4  minutes. 
=    1  hour. 
==    2  hours. 
=    6  hours. 
=  24  hours. 

40.  Every  circle  is  supposed  to  be  divided  into  360  equal 
parts  called  degrees,  each  degree  into  60  equal  parts  called 
minutes,  and  each  minute  into  60  equal  parts  called  seconds. 
For  astronomical  purposes,  the  circumference  of  the  circle  is 
also  supposed  to  be  divided  into  12  equal  parts,  each  of  which 

Quest. — How  do  you  find  the  number  of  days  from  the  fourth  of  March 
to  the  fifteenth  of  August  ?  What  is  the  number  of  days  from  the  tenth  of 
October  to  the  third  of  June?  Also  the  same  in  a  leap  year?  40.  How 
is  any  circle  supposed  to  be  divided  ?  What  is  a  sign,  or  sign  of  the  zodiac  ? 
Repeat  the  table. 


OP   THE    DENOMINATION    OF    NUMBERS.  39 

is  called  a  sign.    The  characters  which  mark  these  divisions 
are  as  follows : 

c  s  o  '  '' 

circumference,  sign,  degree,  minutes,  seconds, 

TABLE    OF    PARTICULARS. 

41,  For  various  things 

lathings  make  1  dozen. 

12  dozen     -       -       -  -     1  gross. 

12  gross,  or  144  dozen  -     1  great  gross. 

ALSO, 

20  things  make  1  score. 

112  pounds  -       -       -  -  1  quintal  of  fish. 

24  sheets  of  paper    -  -  1  quire. 

20  quires    -       -       -  -  1  ream. 

2  reams    -       -       -  -  1  bundle. 


BOOKS. 

A.  sh  et  folded  in  two  leaves  is  called  a  folio. 

(k 

folded  in  four  leaves          -       a  quarto,  or  4to. 

(( 

folded  in  eight 

[eaves        -       an  octavo,  or  8vo. 

i( 

folded  in  twelve  leaves      -       a  duodecimo,  or  12mo. 

i( 

folded  in  eighteen  leaves  -       an  18mo. 

DIMENSIONS    OF    DRAWING    PAPER. 

Demy, 

1  ft.    n\  in.  by  1  ft.    3i  in. 

Medium, 

1  ft.  10    in.  by  1  ft.    6    in. 

Royal, 

2  ft.    0    in.  by  1  ft.    7    in. 

Super  Royal, 

2  ft.    3    in.  by  1  ft.    7    in.  ' 

Imperial, 

2  ft.    5    in.  by  1  ft.    9i  in. 

Elephant, 

2  ft.    3f  in.  by  1  ft.  lOi  in. 

Columbia, 

2  ft.    9J  in.  by  1  ft.  11    in.* 

Atlas, 

2  ft.    9    in.  by  2  ft.    2    in. 

Double  Elephant, 

3  ft.    4    in.  by  2  ft.    2    in. 

Antiquarian, 

4  ft.    4    in.  by  2  ft.    7    in. 

Quest. — 41.  Repeat  the.  table  of  particulars.    Also  for  books.    What  are 
the  dimensions  of  drawing  paper  ? 


40  OF    THE    FORMATION    OF    NUMBERS. 


REMARKS    ON    THE    FORMATION    OF    NUMBERS. 

42.  We  have  seen  (Art.  10)  that  when  figures  are  written 
by  the  side  of  each  other,  thus, 

8562041304723 
the  language  implies  that  the  unit  in  each  place  is  equal 
to  ten  units  of  the  place  next  to  the  right ;  or  that  ten  units 
of  any  one  place  make  one  unit  of  the  place  next  to  the  left. 

43.  When  figures  are  written  thus, 

£       s.        d.    far. 

4       17      10      3 
the  language  implies,  that  four  units  of  the  lowest  deno;pina- 
tion  make  one  of  the  second ;  twelve  of  the  second,  one  of  the 
third;  and  twenty  of  the  third,  one  of  the  fourth. 

44.  When  figures  are  written  thus, 

T.      cwt.    qr,      Ih.      oz,      dr. 

27       17      2       27       11       10 
the  language  implies,  that  16  units  of  the  lowest  denomination 
make  one  of  the  second ;   16  of  the  second,  one  of  the  third ; 
28  of  the  third  make  one  of  the  fourth  ;  four  of  the  fourth,  one 
of  the  fifth ;  and  20  of  the  fifth,  one  of  the  sixth. 

All  the  other  denominate  numbers  are  formed  on  the  same 
principle  ;  and  in  all  of  them  we  pass  from  a  lower  to  the  next 
higher  denomination  by  considering  how  many  units  of  the 
one  make  one  unit  of  the  other. 

45.  In  our  written  language,  each  of  its  elementary  letters 
has  a  particular  signification,  which  must  be  learned  as  a  first 
step.  We  next  learn  to  place  these  letters  in  the  form  of 
words,  and  then  what  may  be  done  by  using  these  words  in 
connection  with  each  other. 

Quest. — 42.  When  figures  are  written  by  the  side  of  each  other,  what 
does  the  language  imply  ?  43.  AVhen  figures  are  written  with  the  mark 
£>  s.  d.  far.  placed  over  them,  what  does  that  language  imply  ?  44.  When 
figiu-es  are  written  with  T.  cwt.  qr.  lb.  oz.  dr.  placed  over  them,  what  re- 
lation exists  between  the  orders  of  their  units  ?  How  do  we  always  pass 
from  one  denomination  of  denominate  numbers  to  another  ?  45.  How  do 
w©  learn  a  common  language? 


OF    REDUCTION.  41 

So  in  figures :  we  first  learn  what  each  figure  expresses  by 
itself,  and  then  what  it  is  made  to  express  in  all  the  various 
ways  in  which  it  may  be  written.  We  thus  learn  the  lan- 
guage of  figures. 

46.  Let  us  give  a  few  examples  of  the  changes  which  are 
produced  in  the  signification,  by  changing  the  places  of  let- 
ters and  figures. 

In  common  language,  was,     is  a  known  word. 

But  the  same  letters  also  give  saw,     an  instrument. 
Also,  375     expresses,  three  hundred  and  seventy-five  ; 

but  573     expresses,  five  hundred  and  seventy-three. 

It  may  be  well  to  observe  that  the  same  letter  has  the  same 
name,  and  generally  represents  the  same  sound  wherever 
it  may  fall  in  a  word.  So,  likewise,  the  same  figure  always 
expresses  the  same  number  of  units,  wherever  it  may  be 
placed.  Thus,  in  the  example  above :  in  the  first  number, 
5  expresses  five  units  of  the  first  order,  and  3,  three  units 
of  the  third.  In  the  second  number,  5  expresses  ^ve  units  of 
the  third  order,  and  3,  three  units  of  the  first  order.  The 
value  of  the  unit,  however,  always  depends  on  the  place  of 
the  figure. 


OF  REDUCTION. 

47.  Reduction  is  changing  the  denomination  of  a  number 
from  one  unit  to  another,  without  altering  the  value  of  the 
number.  Thus,  if  we  have  2  tens,  and  wish  to  reduce  them 
to  the  denomination  of  units  of  the  first  order,  we  multiply  by 
10,  of  add  one  0  ;  this  gives  20  units  of  the  first  order,  which 
are  equal  to  2  tens. 

Quest. — How  must  we  leam  the  language  of  figures  ?  46.  Give  some 
examples  of  the  changes  in  signification  which  are  produced  by  altering  the 
places  of  letters  and  figures.  Has  the  same  letter  always  the  same  name 
and  sound  ?  Has  the  same  figure  always  the  same  name  ?  Does  it  always 
express  the  same  number  of  units  ?  Does  the  value  of  the  unit  expressed 
remain  the  same?  On  what  does  it  depend?  47.  What  is  reduction? 
How  are  tens  reduced  to  units  of  the  first  order? 


42  OF   REDUCTION. 

If,  on  the  contrary,  we  wish  to  reduce  300  to  units  of  the 
second  oriier,  we  divide  by  10,  and  the  quotient  is  30  units 
of  the  second  order,  which  are  equal  to  300.  Had  we  wished 
to  reduce  to- units  of  the  third  order,  we  should  have  divided 
by  100,  giving  3  for  the  quotient:  hence,  reduction  of  de- 
nominate numbers  is  divided  into  two  parts ; 

1st.  To  reduce  a  number  from  a  higher  denomination  to  a 
lower ;  and 

2d.  To  reduce  a  number  from  a  lower  denomination  to  a 
higher. 

The  first  reduction  is  effected  by  beginning  with  the  number 
in  the  highest  denomination.  Multiply  this  number  by  the  value 
of  its  unit  expressed  in  units  of  the  next  lower  denomination,  and 
add  to  the  product  the  number  in  that  denomination.  Proceed 
in  the  same  manner  through  all  the  denominations  to  the  lowest. 

The  second  reduction  is  effected  thus :  Divide  the  given 
number  by  so  many  as  make  one  of  the  denomination  next 
higher ;  set  aside  the  remainder,  if  any,  and  proceed  in  the  same 
manner  through  all  the  denominations  to  the  highest. 

Thus,  in  the  first,  if  we  wish  to  reduce 

£  s.  d. 
3  14  4 
to  pence,  we  first  multiply  the  £3  by  20,  which  gives  60 
shillings.  We  then  add  14,  making  74  shillings.  We  next 
multiply  by  12,  and  the  product  is  888  pence.  To  this  we 
add  4d.,  and  we  have  892  pence,  which  are  of  the  same 
value  as  £3  14^.  4:d. 

If,  on  the  contrary,  we  wished  to  change  892  pence  to 
pounds,  shillings,  and  pence,  we  should  first  divide  by  12  : 
the  quotient  is  74  shillings,  and  4d.  over.  We  again  divide 
by  20,  and  the  quotient  is  £3,  and  14^.  over:  hence,  the  re- 
sult is  £3  14^.  4:d.,  which  is  equal  to  892  pence. 

Quest. — How  will  you  reduce  units  of  the  first  order  to  those  of  the  sec- 
ond? How  to  those  of  the  third?  To  those  of  the  fourth?  Into  how 
many  parts  is  reduction  of  denominate  numbers  divided  ?  How  do  you 
effect  the  first  reduction  ?     How  do  you  effect  the  second  ? 


OP    REDUCnON. 


43 


The  reductions,  in  all  the  denominate  numbers,  are  made 
in  the  same  manner. 


1.  In  £5  5s,,  how  many- 
shillings,  pence,  and  far- 
things 1 

£         s. 

5  5 

20 

105  shillings. 
12 


EXAMPLES. 

In 


1260  pence. 
4 


5040 


5040    farthings,    how 
pence,    shillings,    and 


many- 
pounds  ? 

4)5040  farthings. 
12)1260  pence. 
2|0)10|5  shillings. 
£5~5s. 


In  this  example,  the  reduc- 
tion is  from  a  less  to  a  greater 
unit. 


Here  the  reduction  is  from 
a  greater  to  a  less  unit. 

2.  In  55  guineas,  how  many  shillings,  pence,  and  far- 
things ? 

3.  Reduce  £54  Us.  d^d.  to  farthings. 

4.  Reduce  £77  11^.  lO^d.  to  halfpence 

5.  Reduce  £94  14^.  8d.  to  pence. 

6.  Reduce  £47  14^.  4d.  to  twopences. 

7.  Reduce  £34  Us.  9d.  to  threepences,  and  to  pence. 

8.  In  £108  11^.  6d.,  how  many  sixpences  ? 

9.  How  many  crowns,  half-crowns,  shillings,  sixpences, 
pence,  and  farthings  are  there  in  £54  ? 

10.  Reduce  £74  13^.  9d.  into  shillings,  threepences,  and 
farthings. 

11.  In  11520  farthings,  how  many  pence,  shillings,  and 
pounds  ? 

12.  In  17880  pence,  how  many  pounds  ? 

13.  Reduce  100000  farthings  into  guineas. 

14.  In  50400  halfpence, , how  many  pounds  ? 

15.  In  12050  shillings,  how  many  crowns  and  pounds  ? 

16.  Reduce    311040   pence   into   shillings,   crowns,  and 
poands. 


44  OF    REDUCTION. 

17.  Reduce  17lb.  5oz.  troy  weight  to  graina. 

18.  Reduce  6720  grains  to  ounces. 

19.  In  14  ingots,  or  bars  of  silver,  each  weighing  27 oz, 
lOpwt.^  how  many  grains  ?     How  many  in  one  ? 

20.  How  many  grains  of  silver  in  4lb.  6oz.  12pwt.  ard 
7grA 

21.  How  many  pounds,  ounces,  pennyweights,  and  grains 
of  gold  in  704121  grains  ? 

22.  How  many  of  each  denomination  in  351262  grains  ? 

.  23.  In  25fb,  apothecaries  weight,  how  many  ounces,  drams, 
scruples,  and  grains  ? 

24.  In  907920  grains,  how  many  ounces  and  pounds  1 

25.  In  151ij  Ij  13  l3  2gr.,  how  many  grains? 

26.  In  174947  grains,  how  many  pounds  ? 

27.  In  16Kj  lOg  13  143  7gr.,  how  many  grains? 

28.  In  12  tons,  avoirdupois  weight,  how  many  pounds  ? 

29.  In  31360ZJ.  of  iron,  how  many  tons  ? 

30.  In  37 5cwt.  292lb.  of  copper,  how  many  pounds  ? 

31.  Reduce  740900o2^.  into  cwts.  and  tons, 

32.  In  9r.  I9cwt.  3qr,  27lb.  lAoz.,  how  many  ounces? 

33.  In  144784060^.,  how  many  tons,  cwt.,  qrs.,   lb.,  oz., 
and  drs.  ? 

34.  In  314  yards  of  cloth,  how  many  nails  ? 

35.  In  576  French  ells,  how  many  yards  ? 

36.  Reduce  97yrls.  3qrs.  to  English  ells. 

37.  In  57  pieces  of  cloth,  each  35  ells  Flemish,  how  many 
ells  English  and  nails  ? 

38.  In  14  bales  of  cloth,  each  17  pieces,  each  piece  56 
ells  Flemish,  how  many  yards,  ells  English,  and  ells  French  ? 

39.  In  471  miles,  long  measure,  how  many  furlongs  and 
poles  ? 

40.  In  123200  yards,  how  many  miles  ? 

41.  In  50  miles,  how  many  yards,  feet,  inches,  and  barley- 
corns ? 

42.  Reduce  37mi.  7fur,  37rd.  Qyd.  5ft,  to  feet. 

43.  How  many  barleycorns  will  reach  round  the  earth, 
each  degree  being  69J  miles  ?  and  how  many  quarters  of 


OF    REDUCTION.  45 

barley  are  contained  in  such  a  number  of  barleycorns,  ad- 
mitting 7922  barleycorns  to  fill  a  pint  ? 

44.  In  77 A.  IR.  14P.,  land  measure,  how  many  perches  ? 

45.  In  17280  perches,  how  many  acres  1 

46.  In  50^.  3i^.  lOP.  dsq.yd.  789sq.ft.,  how  many  square 
feet? 

47.  In  175  square  chains,  how  many  square  rods  ? 

48.  In  14976  perches,  or  square  rods,  how  many  acres  ? 

49.  In  83789263P.,  how  many  square  miles  ? 

50.  In  28  tons  of  round  timber,  how  many  solid  inches'? 

51.  In  155  cords  of  wood,  how  many  solid  feet? 

52.  In  17  cords  of  wood,  how  many  solid  inches  ? 

53.  In  56320  solid  feet,  how  many  cords  ? 

54.  Reduce  349938  cord  feet  to  cords. 

55.  In  32hhds.,  wine  measure,  how  many  quarts  ? 

56.  In  3276  gallons,  how  many  tuns  ? 

57.  In  75hhds.,  how  many  pints  ? 

58.  In  77hhds.  of  brandy,  how  many  half-ankers  ?  - 

59.  In  lOtuns  2hhds.  ISgals.  of  wine,  how  many  gills? 

60.  In  98  hogsheads  of  ale,  how  many  pints  ? 

61.  In  38  butts  of  porter,  how  many  pints  ? 

62.  In  516  barrels  of  beer,  how  many  half-pints  ? 

63.  How  many  gallons  of  beer  are  contained  in  50  barrels  ? 

64.  In  44  quarters  of  corn,  how  many  pecks  ? 

65.  In  30720  quarts,  how  many  lasts  ? 

66.  How  many  sacks  in  103  London  chaldrons  and  13 
bushels  of  coal  ? 

67.  How  many  seconds  in  a  year  of  365da.  6hr,  ? 

68.  How  many  seconds  in  6  years  of  365da,  23hr.  57m. 
39sec.  each? 

69.  In  7569520118  seconds,  how  many  years  of  365  da 
each? 

70.  In  5927040  minutes,  how  many  weeks  ? 


46 


ADDITION. 


ADDITION. 


48.  The  sum  of  two  or  more  numbers,  is  a  number  which 
contains  as  many  units,  and  no  more,  as  are  found  in  all  the 
numbers  added;  and 

Addition  is  the  process  of  finding  the  sum  or  sum  total 
of  two  or  more  numbers. 

If  3  be  added  to  5  their  sum  will  be  8,  and  the  unit  of  the 
number  8  will  be  the  same  as  the  unit  of  the  numbers  3  and  5. 
The  numbers  3  and  5,  which  are  thus  added,  must  have  the 
same  unit ;  for,  if  3  denoted  tens,  and  5  expressed  units  of 
the  first  order,  their  sum  would  neither  be  8  tens  nor  8  sim- 
ple units.  So  if  3  expressed  yards,  and  5  feet,  their  sum 
would  neither  be  8  yards  nor  8  feet. 

4*9.  Small  numbers  may  be  added  mentally ;  but  it  is  not 
convenient  to  add  large  numbers  without  first  writing  them 
down.     How  are  they  to  be  written  1 

If  we  place  one  above  the  other,  units 
of  the  same  kind  will  fall  in  the  same  ver- 
tical line,  and  the  units  of  the  same  order 
will  fall  directly  under  each  other  in  the 
sum. 

Again,  let  it  be  required  to  add  together  324  and  635. 
In  the  first  number  there  are  4  units,  2 
tens,  and  3  hundreds.  In  the  second,  5 
units,  3  tens,  and  6  hundreds.  Let  the  fig- 
ures of  each  order  of  units  be  placed  under 
those  of  the  same  order,  and  added :  their 
sum  will  be  9  units,  5  tens,  and  9  hundreds, 
or  nine  hundred  and  fifty-nine. 

Quest. — 4:8.  What  is  the  sum  of  two  or  mcSSp.  numbers  ?  What  is 
addition  ?  What  numbers  can  be  blended  into  one  sum  ?  49.  How  may 
small  numbers  be  added  ?    How  are  numbers  written  down  for  addition  ? 


OPERATION. 

3 

Sum    8" 


OPERATION. 

•a 

|g| 

324 

635 

Sum    959 


ADDITION.  -47 

50.  Add  together  the  numbers  894  and  637. 
Write  the  numbers  thus      -       -       -       - 


OPERATION. 

894   • 
637 


« 

11 

.        .        . 

12 

- 

14 

Sum  total 

1531 

And  draw  a  line  beneath  them 
sum  of  the  column  of  units 
sum  of  the  column  of  tens 
Bum  of  the  column  of  hundreds 


In  this  example,  the  sum  of  the  units  is  11,  which  cannot 
be  expressed  by  a  single  figure.  But  11  units  are  equal  to 
1  ten  and  1  unit;  therefore,  we  set  down  1  in  the  place  of 
units,  and  1  in  the  place  of  tens.  The  sum  of  the  tens  is  12. 
But  12  tens  are  equal  to  1  hundred,  and  2  tens ;  so  that  1  is 
set  down  in  the  hundred's  place,  and  2  in  the  ten's  place. 
The  sum  of  the  hundreds  is  14.  The  14  hundreds  are  equal 
to  1  thousand,  and  4  hundreds ;  so  that  4  is  set  down  in  the 
place  of  hundreds,  and  1  in  the  place  of  thousands.  The 
sum  of  these  numbers,  1531,  is  the  sum  sought. 

The  example  may  be  done  in  another  way,  thus : 
Having  set  down  the  numbers,  as  before, 
we  say,  7  and  4  are  1 1  :  we  set  down  1 
in  the  units  place,   and   write  the  1   ten 
under  the  3  in  the  column  of  tens.     We 
then  say,  1  to  3  is  four,  and  9  are  13.    We 
set  down  the  3  in  the  tens  place,  and  write 
the  1  hundred  under  the  6  in  the  column  of  hundreds.     We 
then  add  the  1,  6,  and  8  together,  for  the  hundreds,  and  find 
the  entire  sum  1531,  as  before. 

When  the  siun  in  any  one  of  the  denominations  exceeds 
10,  or  an  exact  number  of  tens,  the  excess  must  be  written 
down,  and  a  number  equal  to  the  number  of  tens  added  to  the 
next  higher  denomination.     This  is  called  carrying  to  the 

Quest.— 50.  What  is  the  sum  of  the  units  ?  What  of  the  tens  ?  What 
of  the  hundreds  ?     What  the  entire  sum  ? 


OPERATION. 
894 

637 

11 

1531 


48 


ADDITION 


ried  may  be  written  under  that  column  or  remembered  and 
added  in  the  mind. 

61.  What  is  the  sum  of  the  numbers  375,  6321,  and  598? 

In  this  example,  the  small  figure  placed 
under  the  4,  shows  how  many  are  to  be 
carried  from  the  first  denomination  to  the 
second,  and  the  small  figure  under  the  9, 
how  many  are  to  be  carried  from  the  sec- 
ond to  the  third.  In  like  manner,  in  the 
examples  below,  the  small  figure  under  each  column  shows 
how  many  are  to  be  carried  to  the  next  higher  denomination. 
Beginners  had  better  set  down  the  numbers  to  be  carried  as 
in  the  examples. 


OPERATION. 

375 

6321 

598 

7294 
11 


(2.) 

96972 

3741 

9299 

(3.) 

9841672 
793139 

888923 

(4.) 

81325 

6784 

2130 

Sum  110012 

2221 

Sum  11523734 

221111 

Sum  90239 

1110 

52.  Let  it  be  required  to  find  the  sum  of  £14  7s.  Sd.  Sfar., 
and  £6  18^.  9d.  2far. 

We  write  down  the  numbers,  as  before,  so  that  units  of 
the  same  value  shall  fall  under  each  other.  Beginning  with 
the  lowest  denomination,  we  find  the 
sum  to  be  5  farthings.  But  as  4  far- 
things make  a  penny,  we  set  down 
the  1  farthing  over,  and  carry  1  to 
the  column  of  pence.  The  sum  of 
the  pence  then  becomes  18,  which 
make  1  shilling  and  6  over.  Set  down  the  6,  and  carry  1  to 
the  column  of  shillings,  the  sum  of  which  becomes  26  ;  that 
is,  1  pound  and  6  shillings.     Setting  down  the  6  shillings  and 


OPERATION. 

£      s.    d.  far. 


14 
6 

7  8  3 
18  9  2 

21 

6  6  1 

Quest. — How  may  the  units  to  bo  carried %e  disposed  of? 

iec^bro 


51.  How 
will  you  remember  how  many  are  to  be  carried^^om  one  column  to  an 
other?  52.  Explain  the  manner  of  adding  pounds,  shillings,  and  pence, 
and  of  passing  from  one  denomination  to  another. 


ADDITIOIV. 


49 


carrying  1  to  the  column  of  pounds,  we  find  the  entire  sum  to 
be  £21  es,  6d.  Ifar. 

53.  Hence,  for  the  addition  of  all  numbers, 

Write  down  the  numbers  so  that  units  of  the  same  denomi- 
nation shall  fall  in  the  same  column,  and  draw  a  line  beneath 
them. 

Add  up  the  units  of  the  lowest  denomination,  and  divide  their 
sum  by  so  many  as  make  one  of  the  denomination  next  higher. 
Set  down  the  remainder  and  carry  the  quotient  to  the  next  higher 
denomination,  and  proceed  in  the  same  manner  through  all  the 
denominations  to  the  last. 


PROOF    OF    ADDITION. 

54.  The  proof  of  an  arithmetical  operation  is  a  second 
operation,  by  means  of  which  the  first  is  shown  to  be  correct. 

Addition  may  be  proved  by  adding  all  the  columns  down- 
ward. It  may  also  be  proved  by  dividing  the  numbers  to  be 
added  into  two  parts,  adding  each  of  the  parts  separately,  and 
then  adding  their  sums.  If  the  last  sum  is  the  same  as  that 
of  all  the  numbers  first  found,  the  work  may  be  considered 
right. 

EXAMPLES. 


182796 

182796      32160 

143274 

143274      47047 

32160 
47047 

Partial  sums  326070     79207 

Sum  405277 

326070  1st  partial  sum. 
79207  2d    " 

Proof  405277 

Quest. — 53.  What  is.  the  general  rule  for  addition?  54.  What  is  the 
proof  of  an  arithmetical  operation  ?  What  is  the  first  method  of  proving 
addition?     What  the  second ? 


50 


ADDITION. 


(1.) 

34578-1 

3750 

87 

328 

17 

327 


Sums  total 
Partial  sums 


4509 


Proofs  39087 


(2.) 
22345 

67890 

8752 

340 

350 

78 

77410 -J 
99755 


(3.) 
234561 
78901 
23456 
78901 
23456 
78901 

283615-^ 
307071 


(4.) 

(5.) 

(6.) 

672981043 

1278976 

8416785413 

67126459 

7654301 

6915123460 

39412767 

876120 

31810213 

7891234 

723456 

7367985 

109126 

31309 

654321 

84172 

4871 

37853 

72120 

978 

2685 

7.  Add  together  six  tens,  fourteen  hundreds,  seven  thoni- 
sands,  nine  ten  thousands,  forty-five  millions,  and  six  thou- 
sand seven  hundred  and  fifty-one. 

8.  What  is  the  sum  of  six  hundreds,  eight  units  of  the  fifth 
order,  thirteen  of  the  sixth,  twenty  of  the  second,  forty  of  the 
third,  and  two  billions,  three  millions,  four  trillions,  two  hun- 
dred and  twenty-one  thousand  seven  hundred  and  fifty-five  ? 

9.  What  is  the  sum  of  eight  hundred  units  of  the  first  or- 
der, sixty  of  the  second,  one  thousand  of  the  third,  ninety-nine 
of  the  fourth,  one  hundred  of  the  fifth,  six  trillions,  one  bil- 
lion, forty-nine  thousand  eleven  hundred  and  sixty-one  ? 

10.  What  is  the  sum  of  three  hundred  and  forty  units  of 
the  third  order,  seven  thousand  six  hundred  and  fifty  of  the 
fourth,  three  millions  of  the  second,  and  six  trillions  seven 
hundred  and  ninety-nine  of  the  first  ? 

11.  Collect  together  into  one  sum,  two  hundred  and  seven- 
ty-eight millions  four  thousand  six  hundred  and  sixty-nine ; 
seventy-six  billions  four  hundred  and  fifty-eight  millions  four 


ADDITION.  51 

hundred  and  seventy-five  ihousand  five  hundred  and  two  ;  fifty- 
billions  three  hundred  millions ;  four  hundred  and  seventy- 
tw^o  millions  four  thousand  five  hundred  and  fifty-five ;  nine 
millions  seven  hundred  thousand  three  hundred  and  two ; 
twelve  millions  three  hundred  thousand  four  hundred  and 
sixty-one ;  two  hundred  millions  four  hundred  thousand  and 
four;  eight  hundred  millions  seven  hundred  and  forty-nine 
thousand  seven  hundred  and  ninety-nine ;  two  hundred  and 
six  millions  four  hundred  and  forty  thousand  and  thirty-four. 

12.  Find  the  sum  total  of  five  billions  six  hundred  and 
forty-nine  millions  three  hundred  and  seven  thousand  and 
sixty;  nine  hundred  and  forty  millions  three  hundred  and 
seventy-four  thousand  six  hundred  and  eighty-one  ;  nine  bil- 
lions eight  hundred  and  seventy- six  piillions  five  hundred  and 
forty-three  thousand  two  hundred  and  ten ;  one  hundred  and 
twenty-three  millions  four  hundred  and  fifty-six  thousand 
seven  hundred  eighty-nine  ;  five  billions  three  hundred  mil- 
lions seven  hundred  and  seventy-seven  thousand  seven  hun- 
dred and  seven. 

13.  Add  together  seven  hundred  and  four  billions  three 
hundred  and  sixty-millions  five  hundred  and  thirteen  thousand 
and  forty-two  ;  sixty-four  billions  seven  hundred  and  ninety- 
three  millions  six  hundred  and  twenty-nine  thousand  five 
hundred  and  forty-eight ;  six  hundred  and  ninety-nine  billions 
six  hundred  and  ninety-nine  millions  eight  hundred  and  sixty- 
five  thousand  seven  hundred  and  seventy -five. 

14.  Collect  together  and  find  the  sum  of  fifty-eight  billions 
nine  hundred  and  eighty-two  millions  four  hundred  and  eighty- 
seven  thousand  six  hundred  and  fifty-four ;  seven  hundred 
and  forty  billions  three  hundred  and  fifty  millions  five  him- 
dred  and  forty  thousand  seven  hundred  and  sixty ;  four  hun- 
dred and  twenty-five  billions  seven  hundred  and  three  millions 
four  hundred  and  two  thousand  six  hundred  and  three  ;  thirty- 
four  billions  twenty  millions  forty  thousand  and  twenty ;  five 
hundred  and  sixty  billions  eight  hundred  millions  seven  hun- 
dred thousand  and  four  hundred 


52 


ADDITION. 


(15.) 

$87,046 

21,846 


(16.) 

$950,60 

107,27 


(17.) 

$109,049 

691,027 


(18.) 

$8704,067 

7504,61 


19.  What  is  the  sum  of  6  eagles  15  dollars  75  cents  5  mills, 
+  4  eagles  100  dollars  30  cents  8  mills,  +  607  dollars  8  cents 
1  mill,  +  407  eagles  604  dollars  89  cents  9  mills  ? 

20.  What  is  the  sum  of  47  eagles  207  dollars  51  cents 
8  mills,  +  4  eagles  49  dollars  1  cent  1  mill,  +  1000  eagles 
40009  dollars  16  cents  9  mills,  +  691  eagles  9791  dollars 
14  cents  2  mills  ? 


(21.) 

(22.) 

(23.) 

(24.) 

£,     s,     d. 

£,    s.     df. 

£  s.     d. 

£    s.     d. 

149  14  1\ 

14  11  3i 

14  19  41 

14  10  4^ 

37  11  9f 

19  18  10 

17  11  10 

77  18  3 

64  14  7 

77  11  3i 

39  18 -Hi 

14  13  9^ 

104  19  Hi 

49  14  7 

19  14  9 

67  12  4| 

64  13  10 

16  18  41 

19  15  111 

9  11  10 

174  19  llf 

17  15  10 

18  19  10 

18  10  5 

47  14  lOi 

1  14  9i 

77  19  Hi 

17  19  4 

39  15  11^ 

6  18  lOf 

14  11  lOi 

19  10  4 

7^0   A  ^^ 

(25.) 

(26.)  ' 

(27.) 

(28.) 

Ih,    oz.pwt. 

oz.  pwt.  gr. 

lb,  oz.  pwt. 

oz.pwt.  gi , 

174  11  19 

174  19  23 

71  11  19 

74  19  23 

74  10  13 

714  11  14 

64  8  14 

64  14  17 

944  9  14 

714  0  18 

77  0  0 

74  19  11 

74  11  19 

74  1  22 

14  3  11 

QQ   13  9 

944  10  13 

948  2  21 

64  2  9 

74  14  11 

74  11  3 

74  1  12 

74  1  14 

14  10  3 

14  9  4 

715  2  14 

77  2  13 

19  11  14 

77  10  11 

714  18  16 

19  2  14 

17  10  13 

ADDITION. 

53 

(29.) 

(30.) 

(31.) 

(32.) 

^    5 

3 

5 

3 

3 

3   3  gr. 

ft  f   3 

47  11 

7 

149 

7 

2 

749  2  19 

84  11  7 

94  10 

6 

714 

3 

0 

607  1  18 

74  10  6 

74  10 

4 

619 

2 

1 

714  2  17 

37  5  4 

75  9 

3 

74 

6 

2 

400  0  0 

19  4  3 

69  0 

2 

169 

5 

2 

74  1  13 

74  1  2 

57  1 

2 

74 

1 

2 

715  2  14 

79  2  6 

18  2 

1 

777 

6 

1 

64  1  18 

19  2  4 

74  1 

2 

948 

5 

2 

174  2  19 

74  a.  5 

(33.) 

(34.) 

(35.) 

(36.) 

T.  cwt. 

qr. 

cwt. 

gr. 

Z&. 

qr.  lb.  oz. 

lb.  oz.  dr. 

174  19 

3 

174 

3 

27 

44  27  15 

17  15  15 

74  14 

2 

714 

2 

24 

74  26  14 

27  14  11 

714  13 

1 

149 

1 

14 

19  14  13 

16  13  9 

718  16 

2 

719 

2 

16 

74  19  14 

74  14  14 

734  15 

2 

407 

1 

23 

66  27  13 

70  0  0 

714  14 

1 

149 

2 

17 

74  19  10 

64  13  10 

70  13 

2 

714 

2 

18 

14  18  11 

74  14  11 

(37.)  (38.)  (39.)  (40.) 

yd.  qr.  na.      E.E.  qr.  na.      E.  Fr.  qr.  na.     E.  Fl.  qr.  na. 


74 

3 

3 

77 

4 

3 

749 

5 

3 

714 

2 

3 

64 

2 

1 

14 

3 

2 

704 

4 

2 

615 

1 

2 

74 

1 

3 

■  74 

2 

1 

108 

3 

1 

714 

1 

3 

49 

2 

1 

49 

1 

2 

705 

4 

0 

724 

2 

2 

74 

1 

2 

74 

2 

1 

708 

3 

1 

149 

1 

2 

44 

3 

1 

74 

3 

2 

474 

5 

2 

718 

2 

3 

74 

2 

0 

44 

1 

2 

174 

0 

1 

419 

1 

1 

14 

1 

2 

74 

2 

3 

194 

3 

2 

710 

1 

2 

54  ADDITION. 

(41.)       (42.)         (43.)  (44.) 

L,  mi.  fur.  Fur.  rd.  yd.  Rd.  yd.  ft  Ft.  in.  bar 

17  2  7  147  39  5\  177  5i  2  174  11  2 

14  1  6  614  37  4f  714  4J  1  49  10  1 

74  1  7  714  19  3^  714  1^  2  74  11  2 

69  2  4  674  17  1^  615  0  1  64  9  1 

74  1  0  719  27  2|  714  If  2  74  10  1 

69  2  1  '  197  19  1^  719  1^  1  64  11  2 

74  1  2  714  14  3i  437  2f  1  74  10  0 

94  0  3  704  19  4|  614  1^  2  64  9  1 


(45.) 
A.  R.  P. 

(46.) 
A.    R. 

P. 

(47.; 
A.  R. 

) 
P. 

(48.) 
A.    R. 

P. 

77  3  39 

714  3 

39 

14  3 

39 

174  3 

39 

64  2  37 

619  1 

18 

74  1 

19 

714  1 

27 

74  1  24 

714  2 

27 

64  2 

14 

618  2 

12 

64  2  19 

619  1 

34 

74  1 

18 

719  1 

14 

74  1  18 

719  2 

37 

47  2 

24 

734  2 

11 

64  2  17 

719  1 

24 

18  1 

14 

715  1 

24 

14  1  13 

615  2 

14 

74  2 

19 

639  2 

14 

74  2  11 

74  1 

18 

34  1 

14 

714  3 

24 

(49.) 

(50.) 

1 

(51.) 

(52.) 

Tun  hhd. 

gal 

Pun.  gal.  qt. 

Tierce  gal. 

qt. 

Gal.  qt.pt. 

714  3 

62 

714  83 

3 

74 

41 

3 

14  3  1 

614  2 

61 

615  81 

2 

64 

40 

2 

74  2  1 

174  1 

39 

714  74 

1 

74 

19 

1 

39  2  1 

164  2 

47 

614  18 

2 

64 

39 

.2 

17  1  0 

274  1 

49 

713  75 

0 

74 

40 

1 

19  2  0 

175  2 

37 

614  17 

] 

69 

Id 

1 

77  1  1 

375  1 

49 

715  14 

3 

17 

39 

2 

39  3  1 

714  2 

61 

719  28 

2 

18 

41 

1 

14  1  1 

ADDITION. 

55 

(53.) 

(54.) 

(55.) 

(56.) 

Bar.fi  . 

gal. 

Bar.  fir.  gal. 

Hhd.  gal. 

qt. 

HU. 

gal. 

qU 

74  3 

8 

73  3  7 

714  47 

3 

714 

53 

3 

14  2 

7 

69  2  6 

6JL4  44 

1 

415 

47 

2 

16  1 

4 

14  1  7 

374  43 

2 

714 

19 

1 

17  1 

3 

39  2  2 

157  41 

1 

614 

27 

1 

29  2 

2 

19  1  6 

719  42 

1 

715 

51 

2 

17  1 

7 

49  2  6 

374  41 

2 

714 

37 

2 

41  2 

6 

37  1  4 

174  12 

1 

615 

19 

1 

37  1 

5 

19  1  2 

19  13 

2 

714 

18 

2 

(570 

1 

(58.) 

(59.; 

\ 

(60.) 

It.ch.  bu. 

pk. 

Weys 

qr.  bu. 

Qr.   bu. 

pk. 

Scows.  L.ch.  bu 

14  31 

3 

174 

3  7 

149  7 

3 

74 

20  35 

74  31 

2 

375 

1   6 

715  3 

2 

49 

19  33 

64  30 

1 

400 

0  5 

649  1 

3 

64 

17  35 

74  27 

2 

371 

1  4 

479  2 

1 

74 

14  10 

64  19 

2 

634 

2  3 

675  1 

3 

39 

13   9 

74  31 

1 

719 

1  2 

149  2 

1 

47 

16   3 

64  11 

1 

149 

2  1 

375  1 

2 

19 

17   4 

95  10 

2 

375 

1  3 

649  1 

3 

37 

18  34 

(61.) 

(62.) 

(63.) 

(64.) 

Yr.   mo. 

wk. 

Mo.  wk 

.da. 

Da.    hr.  min. 

Hr. 

min.  sec. 

737  12 

3 

64  3 

6 

714  23 

59 

647 

59  59 

347  11 

2 

74  1 

5 

74  14 

54 

137 

54  54 

618  10 

1 

34  2 

3 

64  21 

55 

375 

56  56 

374   9 

2 

74  1 

4 

74  13 

53 

714 

17  19 

175   3 

1 

63  2 

1 

69  12 

14 

615 

54  54 

714  12 

3 

74'  1 

2 

74  12 

19 

714 

17  13 

615  10 

1 

64  2 

1 

37  11 

17 

613 

34  56 

714   3 

1 

74  1 

3 

16  12 

19 

624 

27  39 

66  ADDITION. 


APPLICATIONS. 


1.  In  1843,  the  number  of  acres  of  the  public  lands  sold 
in  the  several  states  and  territories  was  as  follows : — In 
Ohio,  13338  acres,  Indiana  50545,  Illinois  409767,  Mis- 
souri 436241,  Alabama  178228,  Mississippi  34500,  Louisi- 
ana 102986,  Michigan  12594,  Arkansas  47622,  Wisconsin 
167746,  Iowa  143375,  Florida  8318.  What  was  the  whole 
number  of  acres  sold  in  the  United  States  ? 

2.  The  number  of  acres  of  the  public  lands  sold  in  1834 
was  4658218  ;  in  1835,  12564478  ;  in  1836,  25167833.  The 
number  sold  in  1840  was  2236889;  in  1841,  1164*^96;  in 
1842,  1129217.  How  many  acres  were  sold  in  the  first 
three,  and  how  many  in  the  last  three  years  ? 

3.  In  1844,  the  school  fund  of  Connecticut  was  invested 
as  follows  :  in  bonds  and  mortgages,  $1695407,44  ;  in  bank 
stock,  $221700  ;  in  cultivated  lands,  and  buildings,  $78367  ; 
in  wild  lands,  $52493,75  ;  in  stock  in  Massachusetts,  $210  ; 
in  cash,  $3245,58.  What  was  the  whole  amount  of  the 
fund? 

4.  The  salaries  of  the  English  cabinet  ministers  are  as 
follows :  of  the  First  Lord  of  the  Treasury,  £5000 ;  of  the 
Lord  High  Chancellor,  £14000 ;  of  the  Lord  President  of 
the  Council,  £2000  ;  of  the  Lord  Privy  Seal,  £2000  ;  of  the 
Secretaries  of  State  for  the  Home,  Foreign,  and  Colonial 
Departments,  £15000  ;  of  the  Chancellor  of  the  Exchequer, 
£5000 ;  of  the  First  Lord  of  the  Admiralty,  £4500 ;  of  the 
Paymaster-general,  £2500 ;  of  the  President  of  the  Board 
of  Control,  £2000.  Required  the  sum  of  the  salaries  of  the 
cabinet. 

5.  What  was  the  whole  number  of  pieces  coined  in  the 
United  States' mint  in  1835,  there  having  been  371534  half- 
eagles,  131402  quarter-eagles,  5352006  half-dollars,  1952000 
quarter-dollars,  1410000  dimes,  2760000  half-dimes,  3878000 
cents,  and  141000  half-cents?     Required  also  the  value  of 

he  whole  number  of  coins  executed  in  that  year. 

6.  The  value  of  the  imports  during  Mr.  Monroe's  second 


ADDITION.  57 

administration  was,  in  1821,  $62585724;  in  1822,$83241541 ; 
in  1823,  $77579267;  in  1824,  $80549007.  The  value  of 
the  exports  in  1821,  was  $64974382  ;  in  1822,  $72160281  ; 
in  1823,  $74699030;  in  1824,  $75986657.  What  was  the 
amount  of  imports  and  the  amount  of  exports  in  that  term  ? 

7.  What  was  the  population  of  the  British  provinces  in 
North  America  in  1834,  the  population  of  Lower  Canada 
being  stated  at  549005,  of  Upper  Canada  336461,  of  New 
Brunswick  152156,  of  Nova  Scotia  and  Cape  Breton  142548 
of  Prince  Edward^s  Island  32292,  of  Newfoundland  75000  ? 

8.  What  was  the  population  of  Brazil  in  1819,  there  having 
been  of  whites  843000 ;  of  free  people  of  mixed  blood,  426000 ; 
of  Indians,  259400  ;  of  free  negroes,  159500  ;  of  negro  slaves^ 
1728000  ;  of  slaves  of  mixed  blood,  202000  ? 

9.  The  imports  into  France,  in  1826,  were  valued  at 
564728392  francs;  in  1827,  at  565804228  francs;  in  1828, 
607677321  francs;  in  1829,  616353397  francs;  in  1830, 
638338433  francs;  in  1831,  512825551  francs;  in  1832, 
652872341  francs;  in  1833,  693275752  francs.  What  was 
the  value  of  the  imports  for  those  years  ? 

10.  The  number  of  emigrants  in  1837,  from  Great  Britain 
to  British  North  America,  was :  from  England,  5027 ;  from 
Scotland,  2394 ;  and  from  Ireland,  22463.  The  number  to 
the  United  States  the  same  year  was,  from  England,  31769 ; 
from  Scotland,  1130  ;  from  Ireland, 33871.  Required  the  num- 
ber of  emigrants  to  each  place,  and  the  entire  number. 

11.  The  consumption  of  coffee  in  Great  Britain  is  stated 
to  be  10500  tons  a  year;  in  the  Netherlands  and  Holland, 
40500  tons  ;  in  Germany  and  the  countries  round  the  Baltic, 
32000  tons  ;  in  France,  Spain,  Italy,  Turkey  in  Europe,  and 
the  Levant,  35000  tons ;  in  America,  20500  tons.  What  is 
the  entire  consumption  of  coffee  in  these  countries  ? 

12.  The  numbei  of  regular  troops  furnished  by  each  of  the 
states  in  the  revolution,  was  as  follows :  New  Hampshire, 
12497;  Massachusetts,  67907;  Rhode  Island,  5908;  Con 
necticut,  31939;   New  York,   17781;    New  Jersey,  10726 ; 

3* 


58 


ADDITION. 


Pennsylvania,  25678  ;  Delaware,  2386;  Maryland,  13912  ; 
Virginia,  26678;  North  Carolina,  7263;  South  Carolina, 
6417 ;  Georgia,  2679.  What  was  the  number  of  regular 
troops  engaged  during  the  war  ? 

13.  The  revenue  of  the  post-ofSce  at  Albany,  for  the  fourth 
quarter  of  1845,  was  $2697;  at  Baltimore,  $10339;  at 
Brooklyn,  N.  Y.,  $1279  ;  at  Bangor,  Me.,  $1107  ;  at  Buffalo, 
$2339;  at  Cincinnati,  $6103;  at  Detroit,  $1007;  at  Hart- 
ford, $1239;  at  Louisville,  $1946;  at  Mobile,  $4199;  at 
Nashville,  $1194  ;  at  Newark,  N.  J.,  $1026  ;  at  Norfolk,  Va., 
$1175  ;  at  Petersburg,  Ya.,  $1090  ;  at  Philadelphia,  $21642  ; 
at  Pittsburg,  Pa.,  $3612  ;  at  Providence,  $3046  ;  at  Roches- 
ter, N.  Y.,  $2606;  at  Springfield,  Mass.,'$1031  ;  at  Troy, 
N.  Y.,  $1883.  What  was  the  total  amount  of  revenue  received 
from  these  post-offices  ? 

14.  The  list  of  vessels  in  the  British  navy,  on  the  1st  of  Jan 
uary,  1846,  was  as  follows:  sailing  vessels  in  'commission' 
and  in  '  ordinary,'  36 1  ;  sail  vessels  building,  42  ;  steam 
frigates,  11  ;  steam  frigates  building,  12  ;  other  steam  vessels, 
88  ;  steam  vessels  building,  8  ;  steam  packets,  25  ;  receiving 
and  quarantine  vessels,  transports,  &c.,  134.  What  is  the 
whole  number  of  vessels,  and  what  the  number  of  each  kind  ? 

15.  The  deposites  of  gold  for  coinage  at  the  mint  in  Phila- 
delphia, in  1842,  were:  from  mines  in  the  United  States, 
$273587  ;  coins  of  the  United  States,  old  standard,  $27124  ; 
foreign  coins,  $497575  ;  foreign  bullion,  $158780  ;  jewellery, 
$20845.  The  deposites  of  silver  were  :  bullion  from  North 
Carolina,  $6455  ;  foreign  bullion,  $153527  ;  Mexican  dollars, 
$1085374;  South  American  dollars, $2 6372;  European  coins, 
$272282  ;  plate,  $23410.  What  was  the  amount  of  gold  de- 
posited ?     What  of  silver  ?    And  what  the  entire  sum  ? 

16.  Of  the  public  lands,  there  were  ceded  by  the  states 
of  Virginia,  New  York,  Massachusetts,  and  Connecticut, 
169609819  acres;  by  Georgia,  58898522  acres;  by  North 
and  South  Carolina,  26432000  acres  ;  and  987852332  acres 
were  purchased  of  France  and  Spain.  Required  the  number 
of  acres  ceded  and  purchased. 


ABBITION. 


59 


1? .  The  population  of  New  York  city  in  1840  was  312710  ; 
of  Philadelphia,  258037  ;  of  Baltimore,  134379  ;  of  New  Or- 
leans, 102193  ;  of  Boston,  93383  ;  of  Cincinnati,  46338  ;  of 
Brooklyn,  36233  ;  of  Albany,  33721  ;  of  Charleston,  29261  ; 
of  Louisville,  21210;  of  Richmond,  20153;  of  St.  Louis, 
16469,  What  was  the  whol^  number  of  inhabitants  in  these 
twelve  cities  ? 

18.  The  following  table  exhibits  the  population  of  the  sev- 
eral states  and  territories,  at  the  taking  of  each  census  to 
1840.  What  was  the  population  of  the  United  States  in  each 
of  those  years  ? 


States. 

1790. 

1800. 

1810. 

1820. 

1830. 

1840. 

Maine  -  -  - 

96540151719 

228705 

298335 

399955 

501793 

New  Hampshire 

141899;183762 

214360 

244161 

269328 

284574 

Vermont 

85416154465 

217713 

235764 

280652 

291948 

Massachusetts 

378717423245 

472040 

523287 

610408 

737699 

Rhode  Island  - 

69110  69122 

77031 

83059 

97199 

108830 

Connecticut  - 

238141  251002262042 

275202 

297665 

309978 

New  York 

340120'586756  959949 

1372812 

1918608 

2428921 

New  Jersey  - 

184139  211949  249555 

277575 

320823 

373303 

Pennsylvania  - 

434373  602365  810091 

1049458 

1348233 

1724033 

Delaware  -  - 

59098  64273  72674 

72749 

76748 

78085 

Maryland  -  - 

319728  341548  380546 

407350 

447040 

470019 

Virginia   -  - 

748308  880200  974642 

1065379 

1211405 

1239797 

North  Carolina 

393751478103 

555500 

638829 

737987 

753419 

South  Carolina 

249073  345591 

415115 

502741 

581185 

594398 

Georgia   -  - 

82548162101 

252433 

340987 

516823 

691392 

Alabama  -  - 

- 

- 

20845 

127901 

309527 

590756 

Mississippi 

- 

8850 

40352 

75448 

136621 

375651 

Louisiana  -  - 

- 

76556 

153407 

215739 

352411 

Arkansas  -  - 

- 

_ 

- 

14273 

30388 

97574 

Tennessee  -  - 

30791 

105602 

261727 

422813 

681904 

829210 

Kentucky  -  - 

73077 

220955 

406511 

564317 

687917 

779828 

Ohio   -  -  - 

- 

45365 

230760 

581434 

937903 

1519467 

Michigan  -  - 

- 

- 

4762 

8896 

31639 

212267 

Indiana   -  - 

- 

4875 

24520 

147178 

343031 

685866 

Illinois  -  -  - 

- 

- 

12282 

55211 

157455 

476183 

Missouri  -  - 

- 

_ 

20845 

66586 

140445 

383702 

Dist.  Columbia 

_ 

14093  24023 

33039 

39834 

43712 

Florida   -  - 

_ 

_ 

_ 

_ 

34730 

54477 

Wisconsin  -  - 

_ 

_ 

_ 

— 

_ 

30945 

Iowa   -  -  - 

- 

- 

- 

- 

- 

43112 

60 


ADDITION. 


19.  The  slave  population  of  the  states  and  territories,  ac- 
cording to  each  census,  is  shown  in  the  following  table. 
Required  the  number  of  slaves  in  the  United  States  at  each 
enumeration. 


States. 

1790. 

1800. 

1810. 

1820. 

1830. 

1840. 

Maine      -       -       - 

0 

0 

0 

0 

0 

0 

New  Hampshire  - 

158 

8 

0 

0 

0 

1 

Vermont 

17 

0 

0 

0 

0 

0 

Massachusetts 

0 

0 

0 

0 

0 

0 

Rhode  Islands 

952 

381 

103 

48 

17 

5 

Connecticut   - 

2759 

951 

310 

97 

25 

17 

New  York     - 

21324 

20343 

15017 

10088 

75 

4 

New  Jersey  - 

11423 

12422 

10851 

7657 

2254 

674 

Pennsylvania 

3737 

1706 

795 

211 

403 

64 

Delaware 

8887 

6153 

4177 

4509 

3292 

2605 

Maryland 

103036 

105635 

111502 

107398 

102294 

89737 

Virginia 

203427 

345796 

392518 

425153 

469757 

448987 

North  Carolina     - 

100572 

133296 

168824 

295017 

235601 

245817 

South  Carolina 

107094 

146151 

196365 

258475 

315401 

327038 

Georgia  -       -       - 

29264 

59404 

105218 

149656 

217531 

280944 

Alabama 

- 

- 

- 

41879 

117549 

253532 

Mississippi     - 

_ 

3489 

17088 

32814 

65659 

195211 

Louisiana 

_ 

- 

34660 

69064 

109588 

168452 

Arkansas 

_ 

- 

- 

1617 

4576 

19935 

Tennessee     - 

3417 

13584 

44535 

80107 

141603 

183059 

Kentucky 

11830 

40343 

80561 

126732 

165213 

182258 

Ohio        -       -       - 

— 

— 

— 

— 

3 

Michigan 

- 

- 

24 

- 

32 

0 

Indiana    -       -       - 

~ 

135 

.   237 

190 

0 

3 

Illinois    -       -       - 

_ 

_ 

168 

117 

#747 

331 

Missouri 

_ 

- 

3011 

10222 

25081 

58240 

Dist.  Columbia     - 

_ 

3244 

5395 

6377 

6119 

4694 

Florida    -       .       - 

_ 

— 

_ 

- 

15501 

25717 

Wisconsin 

_ 

_ 

_ 

- 

- 

11 

Iowa       -       -       - 

- 

- 

- 

- 

- 

16 

•  Indented  colored  servants. 


SUBTRACTION. 


61 


SUBTRACTION. 

55.  Subtraction  is  the  process  of  finding  the  difference 
between  two  numbers. 

When  the  numbers  are  unequal,  the  larger  of  the  two  is 
called  the  minuend^  and  the  less  is  called  the  subtrahend^  and 
their  difference,  whether  they  are  equal  or  unequal,  is  called 
the  remainder. 

When  the  numbers  are  small,  their  difference  is  apparent, 
and  the  subtraction  may  be  made  mentally. 


EXAMPLES. 

1.  From  869  subtract  327  :  that  is,  from  8  hundreds  6  tens 
and  9  units,  it  is  required  to  take  3  hundreds  2  tens  and  7 
unit's. 

We  begin  at  the  right  hand  figure 
of  the  lower  line,  and  say,  7  from  9 
leaves  2  :  set  down  the  2  under  the  7. 
Proceeding  to  the  next  column,  we 
say,  2  from  6  leaves  4 ;  set  down  the 
4,  and  then  say,  3  from  8  leaves  5.  Hence,  the  remainder 
or  true  difference  between  the  numbers  is  542. 

2.  From  843  subtract  562. 

Beginning  with  the  lowest  denomina- 
tion, we  say,  2  from  3  leaves  1.  At  the 
next  step  we  meet  a  difficulty,  for  we  can- 
not subtract  6  from  4.  If,  now,  we  add  10 
tens  to  the  4,  (which  are  written  in  small 
figures  above,)  and  10  tens  to  the  6  directly  under  it,  it  is 
plain  that  the  difference  will  not  be  affected,  since  both  the 

Quest. — 55.  What  is  subtraction  ?  How  many  numbers  are  considered 
in  subtraction?  What  are  they  called?  When  the  numbers  are  small, 
how  may  the  subtraction  be  made  ?  Ex.  2.  How  do  we  get  over  the  diffi- 
culty in  subtracting  the  tens  ?  If  equal  numbers  be  added  to  the  minuend 
and  subtrahend,  will  their  difference  be  changed  ? 


OPERATION. 
869   Minuend 
327   Subtrahend. 
542   Remainder, 


OPERATION. 

10 

843 
562 


281 


62  SUBTRACTION. 

numbers  are  equally  increased.  But  adding  10  tens  to  the 
6  is  the  same  thing  as  adding  1  to  the  5  hundreds  :  hence, 
we  may  consider  10  to  he  added  to  any  figure  of  the  minuend^ 
provided  we  add  1  to  tlit  next  figure  of  the  subtrahend  to  the 
left. 

We  can  now  go  on  with  the  subtraction ;  for  we  say, 
6  from  14  leaves  8.  Then,  1  carried  to  5  makes  6  :  and  6 
from  8  leaves  2.  Hence  the  remainder  is  281  ;  and  all  simi- 
lar examples  are  done  in  the  same  manner. 


From 

i'. 
6 

cwt. 

20 

14 

qr. 

2 

lb. 

28 

20 

oz. 
12 

take 

4 

1 

17 

1 
1 

21 

10 

emaindei 

:     1 

17 

0 

27 

2 

In  this  example  we  say,  10  ounces  from  12  leaves  2. 
At  the  next  denomination  we  meet  a  difficulty,  for  we  can- 
not subtract  21  from  20.  We  add  to  the'  20  so  many  units 
as  make  1  unit  of  the  next  higher  denomination — that  is,  28, 
and  suppose  at  the  same  time  1  unit  to  be  added  to  that  de- 
nomination in  the  subtrahend.  We  then  say,  21  from  48 
leaves  27 :  then  2  from  2  leaves  0.  In  the  hundreds  we 
again  have  to  add,  after  which  we  say,  17  from  34  leaves 
17 ;  then  we  take  5  from  6,  and  have  the  true  remainder. 

56.  Hence,  to  find  the  difference  between  two  numbers  : 

Set  down  the  less  number  under  the  greater,  so  that  units  of 
the  same  denomination  shall  fall  in  the  same  column,  and  be- 
ginning with  the  lowest  denomination,  subtract  each  from  the 
one  above  it.  When  the  units  in  any  one  denomination  of  the 
subtrahend  exceed  those  of  the  same  denomination  in  the  minu- 
end, suppose  so  many  units  added  in  the  minuend  as  make  one 
unit  of  the  next  higher  denomination ;  after  which  add  one  to 
the  next  denomination  of  the  subtrahend,  and  subtract  as  before. 

Quest. — If  you  add  10  to  any  figure  of  the  minuend,  what  will  you  add 
to  the  subtrahend  ?  Ex.  3.  How  is  the  subtraction  made  in  this  example  ? 
56.  What  is  the  rule  for  subtraction  ? 


SUBTRACTION.  63 


67.  Add  the  remainder  to  the  subtrahend,  and  if  the  sum 
ia  equal  to  the  minuend,  the  work  may  be  regarded  as  right. 
Or,  subtract  the  remainder  from  the  minuend,  and  the  re- 
mainder thus  found  should  be  equal  to  the  subtrahend. 


EXAMPLES. 

From 

(1.) 

10  10 

87407 

(2.) 

10  10  10  10 

27431 

(30 
■  £14    I6s. 

12  4 

take 

6079 

I  1 

19872 
1111 

6     17 
1       1 

9f 

Rem. 

Proof 

4.  From  47348406051320047  take  13456507031079054. 

5.  From  19493899900056075  take  14954298990056076. 

6.  From  500714960079690650  take  742350986470501. 

7.  From  149348761340526465  take  48973024012394. 

(8.)       (9.)      (10.)       (11.) 
From  $374,674    $270,604    $137,04    $9496,004 
take   195,097     191,280     127,97     8496,049 


^Rem. 

12.  What  is  the  difference  between  $487,25  and  $379,674? 

13.  What  is  the  difference  between  $670,04  and  one  hun- 
dred and  four  dollars  and  6  mills  ? 

14.  What  is  the  difference  between  $1000  and  $14,003  ? 

(15.)  (16.)                   (17.)  (18.) 

ih.     oz.  pwt.  oz.  pwt.  gr.  lb.     oz.  pwt.  ^   oz.pwt.gr. 

14     11     9  74  12     13  175     3     10  17    10    20 

11      10  14  64  14     17  159  11      14  14    11    23 


Quest. — 57.  What  is  the  first  method  of  proving  subtraction  ?    What  is 
the  second  ? 


64 

SUBTRACTION. 

(19.) 

(20.) 

(21.) 

(22.) 

^       I 

3 

§   3  3 

3   3  g-r. 

flj   §   3 

]44  10 

5 

27  4  1 

27  1   14 

74  10  5 

64  11 

7 

14  7  2 

14  0  19 

65  11  6 

(23.) 

(24.) 

(25.) 

(26.) 

T.  cwt. 

qr. 

Cwt,  qr,  lb. 

Qr.  Ih.    oz. 

lb.     oz.  dr. 

14  12 

2 

17  1  25 

143  22  12 

174  11  10 

1  14 

3 

14  2  27 

74  19  14 

39  12  13 

(27.)  (28.)  (29.)  (30.) 

Yd.  qr.  na.  E.E.  qr.  na.  E.  Fr.  qr.  na.  E.  Fl.  qr.  na, 

174    2     1  174    3     1  171      1     3  12      1      1 

39    3    2  49    4    2  74     5     2  10     2     3 


(31.) 

(32.) 

(33.) 

(34.) 

L.  mi.  fur. 

Fur.    rd.    yd. 

Rd.  yd.  ft. 

Ft.    in.  bar. 

21  2  4 

13  34  3f 

14  3|  1 

17  11  2 

3  2  6 

12  39  5} 

9  4i  2 

14  11   1 

(35.) 

(36.) 

(37.) 

(38.) 

A.  R.  P. 

A.    R.   P. 

A.  R.  P. 

A.    R.  P. 

12  1  32 

112  1  31 

12  1  25 

19  1  20 

1  3  14 

74  2  37 

10  3  39 

14  2  21 

(39.) 

(40.) 

(41.) 

(42.) 

Tun  hhd.  gal 

Pun.  gal  qt. 

Tier,  gal  qt. 

Gal  qt.ft. 

27  2  54 

147  14  2 

14   1  2 

24  3  0 

19  3   62 

79  83  3 

12  41   3 

17  0  1 

SUBTRACTION. 


65 


(43.) 
Bar,  fir.  gal, 
•14  3  5 
12  3  7 


(44.) 
Bar.  fir.  gal, 
147  1  3 
39  3  8 


(45.) 

Hhd.  gal.  qt. 

271   1  2 

49  47  3 


(46.) 
Hhd.  gal.  qt, 
143  1   2 
79  52  3 


(47.)  (48.) 

L.cA.  hu.  pk.  Weysqr.  hu. 

74  31  3  17  3  1 

47  31  2  14  3  7 


(49.)  (50.) 

Qr.    hu.  ph.  Scows. L.  ch.hu, 

147  6  2  47   1   12 

94  7  3  14  20   35 


(51.)  (52.) 

Yr.  mo.  wk.  Mo.  wk.  da. 

17  11  2  147  2  3 

14  12  3  19  2  4 


(53.) 

Da.    hr.  min. 

167  21  50 

19  23  54 


(54.) 

Hr.  min.  sec. 

147  50  51 

94  59  57 


PROMISCUOUS    EXAMPLES. 

55.  A  horse  in  his  furniture  is  worth  £52  10s. ;  out  of  it, 
j£24  10^.  6d.  How  much  does  the  price  of  the  furniture  ex- 
ceed that  of  the  horse  1 

56.  What  sum  added  to  £11  14^.  d^d.  will  make  £133 
11^.  and9i(^.? 

57.  A  tradesman  failing,  was  indebted  to  A  £105  19^. 
lid.,  to  B  150  guineas,  to  C  £34  18^.  lOd.,  to  D  £500  19^., 
to  E  £700  14^.  9d.  When  this  happened,  he  had  cash  by 
him  to  the  amount  of  £50,  goods  to  the  amount  of  £350  14^. 
9d.,  his  household  furniture  was  worth  £24  11^.,  his  book- 
debts  amounted  to  £94  14^.  8d.  If  these  things  were  faith- 
fully given  up  to  his  creditors,  what  did  they  lose  by  him  1 

58.  The  great  bell  at  Oxford,  the  heaviest  in  England, 
weighs  7T.  llcwt.  3qr.  Alb. ;  St.  PauVs  bell  at  London  weighs 
5T.  2cwt.  Iqr.  22lb. ;  and  Tom  of  Lincoln  weighs  4T.  I6cwt 


66  SUBTRACTION. 

3qr.  I8lb.  How  much  are  these  bells,  together,  inferior  in 
weight  to  the  great  bell  at  Moscow,  the  largest  in  the  world, 
which  weighs  198r.  2cwt.  \qr,  1 

59.  An  apprentice,  who  is  14  years,  11  months,  13  weeks, 
14  hours,  38  minutes  old,  is  to  serve  his  master  till  he  is  21 
years  of  age.     How  long  has  he  to  serve  ? 

60.  What  is  the  difference  of  latitude  and  longitude  be- 
tween Calcutta  in  the  East  Indies,  (lat.  22°  34''  N.,  long. 
88°  34'  E.,)  and  Lima,  in  South  America,  (lat.  12^  V  S., 
long.  760  44^  W.)  ? 

61.  Newton  (Sir  Isaac)  was  born  at  Woolsthorp,  a  ham- 
let in  the  parish  of  Colsworth,  in  Lincolnshire,  on  Sunday,  the 
25th  December,  1642  ;  and  died  at  Kensington,  in  Middlesex, 
on  Monday,  the  20th  March,  1727.  Euler  (Leonard)  was 
born  at  Basil,  in  Switzerland,  on  Tuesday,  the  15th  April, 
1707  ;  and  died  at  Petersburg,  in  Russia,  on  Sunday,  the  7th 
September,  1783.  Lagrange  (Joseph  Louis)  was  born  at 
Turin,  in  Italy,  on  Friday,  the  30th  January,  1736  ;  and  died 
at  Paris,  on  Saturday,  the  10th  April,  1813.  Laplace  (Pierre 
Simon,  marquis  of)  was  born  at  Beaumont-en- Auge,  in  France, 
on  Thursday,  the  23d  March,  1749;  and  died  at  Paris,  on 
Tuesday,  the  27th  March,  1827.  How  old  was  each  of  these 
eminent  philosophers  and  mathematicians  at  the  time  of  his 
decease  ?  and  how  many  years  was  it  from  the  time  each 
died  to  January  1st,  1846. 

62.  In  1840  the  amount  of  tobacco  sent  from  the  United 
States  to  England,  was  26255  hogsheads,  and  to  Holland, 
29534  hogsheads.  How  much  more  was  sent  to  Holland 
than  to  England  ? 

63.  The  population  of  the  northern  district  of  New  York  in 
1840  was  1683068,  and  the  population  of  the  southern  dis- 
trict was  745853.  How  many  more  inhabitants  were  there 
in  the  northern  than  in  the  southern  district,  and  what  was 
the  population  of  the  state  1 

64.  The  population  of  England  in  1841  was  14995508,  the 
population  of  Scotland  2628957,  and  of  Wales  911321.    How 


SUBTRACTION.  67 

much  did  the  population  of  England  and  Wales  combined 
exceed  that  of  Scotland,  and  what  was  the  entire  population 
of  great  Britain  ? 

65.  The  value  of  the  gold  coined  at  the  mint  in  Philadel- 
phia in  1842  was  $960017,50;  the  value  of  that  coined  at 
Charlotte,  N.  C,  was  $159005  ;  at  Dahlonega,  Ga.,  $309648; 
and  at  New  Orleans,  $405500.  How  much  more  was  coined 
at  Philadelphia  than  at  the  three  other  places  1 

66.  The  whole   amount  received  for  the  public  lands  to 

1843,  was  $170940942,62.  There  have  been  paid  for  the 
Indian  title,  the  Florida  and  Louisiana  purchase,  including 
interest,  $68524991,32  ;  and  for  surveying  and  selling,  in- 
cluding salaries  of  officers,  $9966610,14.  Required  the  net 
amount  derived  from  the  sale  of  the  public  lands. 

67.  The  revenue  of  Great  Britain  for  the  year  1843  was 
£50071943,  and  for  the  previous  year,  £44329865.  Re- 
quired the  increase. 

68.  The  value  of  the  merchandise  imported  into  the  Uni- 
ted States  during  the  year  ending  June  30th,  1844,  was 
$108435035  ;  of  which  $24766881  was  admitted  free  of  duty, 
$31352863  paid  specific  duties,  and  the  remainder  paid  du- 
ties ad  valorem.     What  amount  paid  ad  valorem  duties  ? 

69.  The  value  of  the  products  of  the  sea  exported  from  the 
United  States  in  1844,  was  $3350501  ;  the  value  of  the 
products  of  the  forest,  exported  the  same  year,  was  $5808712. 
How  much  more  was  exported  of  the  products  of  the  forest 
than  of  the  sea  ? 

70.  The   imports  from  England  to  the  United  States  in 

1844,  amounted  to  $41476081,  from  Scotland  $527239,  and 
from  Ireland  $88084.  The  value  of  the  exports  to  Eng- 
land, the  same  year,  was  $46940156,  to  Scotland  $1953473, 
and  to  Ireland  $42591.  How  much  did  our  exports  to  Great 
Britain  and  Ireland  exceed  the  imports  ? 

71.  What  was  the  balance  in  the  treasury  of  the  state  of 
Tennessee,  in  October,  1 844,  the  income  for  the  year  ending 


68  SUBTRACTION. 

that  month  having  been  $271823,08,  a  surplus  had  been 
left  the  preceding  year  of  $38875,21  ;  and  the  expenditure 
was  $261416,26? 

72.  The  cost  of  the  internal  improvements  of  the  state  of 
Ohio,  was  $15283783,64,  of  which  the  Ohio  canal  cost 
$4695203,69 ;  the  Miami  canal,  $1237552,16  ;  the  Miami 
Extension,  $2856635,96 ;  and  the  Wabash  and  Erie  canal, 
$3028340,05.  What  was  the  cost  of  the  other  works  of  the 
state  ? 

73.  St.  Augustine  was  founded  Sept.  8th,  1565.  James- 
town was  founded  May  13th,  1607.  The  Battle  of  Prince- 
ton was  fought  Jan.  3d.  1777.  Cornwallis  surrendered,  Oct. 
19th,  1781.  Washington  was  first  inaugurated  April  30th, 
1789:  he  died,  Dec.  14th,  1799.  The  French  Berlin  de- 
cree was  issued  Nov.  21st,  1806,  and  the  British  orders  in 
council,  Nov.  11th,  1807.  The  United  States  declared  war 
against  Great  Britain  June  18th,  1812.  The  Guerriere  was 
captured  by  the  Constitution  Aug.  19th,  1812.  The  frigate 
United  States  captured  the  Macedonian,  Oct.  25th,  1812. 
York  in  Upper  Canada  was  captured  by  the  Americans,  and 
General  Pike  killed,  April  27,  1813.  Fort  George  was  cap- 
tured May  27th,  1813.  The  British  were  repulsed  from 
Sackett's  Harbor  by  the  Americans  commanded  by  General 
Brown,  May  28th,  1813.  The  Battle  of  Lake  Erie  was 
fought  Sept.  10th,  1813.  The  Battle  of  Chippewa  was 
gained  by  a  detachment  of  the  American  army  under  Gen- 
eral Scott,  July  5th,  1814.  The  Battle  of  Niagara,  or  Lundy's 
Lane,  was  fought  July  25th,  1814.  General  Brown  conducted 
the  sortie  from  Fort  Erie,  Sept.  17th,  1814.  The  battle  of 
New  Orleans  was  fought  Jan.  8th,  1815.  Adams  and  Jeffer- 
son died  July  4th,  1826.  The  compromise  bill  was  intro- 
troduced  into  the  senate  Feb.  12th,  1833.  General  Lafayette 
died  May  20th,  1833.  The  Cherokees  began  to  remove  May 
26th,  1838.  What  time  has  elapsed  from  the  dale  of  each  of 
these  events  to  March  17th,  1846? 


MULTIPLICATION. 


MULTIPLICATION. 


68.  If  the  number  1  be  multiplied  by  2,  that  is,  taken  two 
times,  the  result  will  be  2  ;  and  2  is  said  to  be  two  times 
greater  than  1. 

If  1  be  multiplied  by  3,  that  is,  taken  three  times,  the  re- 
sult will  be  3  ;  and  3  is  said  to  be  three  times  greater  than  1. 

If  2  be  multiplied  by  2,  that  is,  taken  2  times,  the  result 
will  be  4  ;  and  4  is  said  to  be  two  times  greater  than  2. 

If  3  be  multiplied  by  4,  the  result  will  be  12  ;  and  12  is 
said  to  he  four  times  greater  than  3. 

In  the  first  case,  1  was  taken  2  times  ;  in  the  second  it 
was  taken  3  times  ;  in  the  third  2  was  taken  2  times  ;  and  in 
the  fourth  3  was  taken  4  times. 

Multiplication  is  a  short  process  of  taking  one  number  as 
many  times  as  there  are  units  in  another.  Hence,  it  is  a  short 
method  of  performing  addition. 

The  number  to  be  taken  is  called  the  multiplicand. 

The  number  denoting  how  many  times  the  multiplicand  is 
to  be  taken,  is  called  the  multiplier. 

The  number  arising  from  taking  the  multiplicand  as  many 
times  as  there  are  units  in  the  multiplier,  is  called  the 
product. 

The  multiplicand  and  multiplier,  together,  are  called  fac- 
tors, or  producers  of  the  product. 

There  are  three  numbers  in  every  multiplication.  First,  the 
multiplicand  ;  second,  the  multiplier ;  and  third,  the  product. 

Quest. — 58.  If  1  be  multiplied  by  2,  what  is  the  result  ?  How  many 
times  greater  is  this  result  than  1  ?  If  3  be  multiplied  by  4,  what  is  the 
result  ?  How  many  times  greater  is  the  result  than  3  ?  What  is  mul- 
tiplication ?  What  is  the  number  to  be  taken  called  ?  What  is  the  number 
showing  how  many  times  the  multiplicand  is  to  be  taken,  called  ?  What 
is  the  result  called?  What  are  the  multiplier  and  multiplicand  taken  to- 
gether called?  How  many  numbei-s  £ire  there  in  every  multiplicatioi  T 
What  are  they  called? 


70  MULTIPLICATION. 

59.  Now,  since  the  product  is  the  result  which  arises  from 
taking  the  multiplicand  as  many  times  as  there  are  units  in 
the  multiplier,  it  follows  that, 

1st.  If  the  multiplier  is  unity,  the  product  will  be  equal  to 
the  multiplicand. 

2d.  If  the  multiplier  contains  several  units,  the  product  will 
be  as  many  times  greater  than  the  multiplicand,  as  the  multi- 
plier is  greater  than  unity. 

3d.  If  the  multiplier  be  less  than  unity,  that  is,  if  it  be  a 
proper  fraction,  then  the  product  will  be  as  many  times  less 
than  the  multiplicand  as  the  multiplier  is  less  than  unity. 

60.  Let  it  be  required  to  multiply  any  two  numbers  to- 
gether, say  6  by  4. 

If  we  make,  in  a  horizontal  line,  as  6 

many  stars  as  there  are  units  in  the 
multiplicand,  and  make  as  many  such 
lines  as  there  are  units  in  the  multi- 
plier, it  is  evident  that  all  the  stars  will 
represent  the  number  of  units  which  re- 
sult frx)m  taking  the  multiplicand  as  many  times  as  there  are 
units  in  the  multiplier. 

Let  us  now  change  the  multiplier  into  the  multiplicand, 
and  let  the  multiplicand  become  the  multiplier.  Then  make, 
in  a  vertical  line,  as  many  stars  as  there  are  units  in  tKe  new 
multiplicand,  and  as  many  vertical  lines  as  there  are  units  in 
the  new  multiplier,  and  it  will  be  again  evident  that  all  the 
stars  will  represent  the  number  of  units  in  the  product. 
Hence, 

Either  of  the  factors  may  he  used  ds  the  multiplier  without 
altering  the  product.     For  example, 

3x7  =  7x3==  21:     also,  6x3  =  3x6  =  18. 

9x5  =  5x9  =  45:     also,  8x6  =  6x8  =  48. 

and, '8x7  =  7x8  =  56:     also,  5x7  =  7x5  =  35. 

Quest. — 59.  If  the  multiplier  is  unity,  how  will  the  product  compare 
with  the  multiplicand  ?  How  will  it  compare  if  the  multiplier  is  greater 
than  unity  1  How  when  it  is  less  ?  60.  If  the  multiplicand  be  made  the 
multiplier,  will  the  product  be  altered  ? 


MULTIPLICATION.  71 

61.  A  composite  number  is  one  that  may  be  produced  by 
the  multiplication  of  two  or  more  numbers,  which  are  called 
the  components  ox  factors.  Thus,  2  X  3  ==  6.  Here  6  is  the 
composite  number,  and  2  and  3  are  the  factors,  or  compo- 
nents. The  number  16  =  8  x  2  :  here  16  is  a  composite 
number,  and  8  and  2  are  the  factors ;  and  since  4x4  =  16, 
we  may  also  regard  4  and  4  as  factors  or  components  of  16. 

Let  it  be  required  to  multiply  8  by  the  composite  number  6, 
of  which  the  factors  are  2  and  3. 

8 


3K     ......     42x8=16  8 

I  (***#****)  

i/-***##***J                    48  24 
3^*     *     *     *     *     #     *     *i 


_2 

48 


If  we  write  6  horizontal  lines  with  8  stars  in  each,  it  is 
evident  that  the  product  of  8  X  6  =  48,  the  number  of  units 
in  all  the  lines. 

But  let  us  first  connect  the  lines  in  sets  of  2  each,  as  on 
the  right ;  there  will  then  be  in  each  set  8  X  2  =  16,  or  16 
units  iy  each  set.  But  there  are  3  sets  ;  hence,  16  X  3  =  48, 
the  number  of  units  in  all  the  sets. 

If  we  divide  the  lines  into  sets  of  3  each,  as  on  the  left,  the 
number  of  units  in  each  set  will  be  equal  to  8X3  =  24,  and 
there  being  2  sets,  24  x  2  =  48,  the  whole  number  of  units. 
As  the  same  may  be  shown  for  any  composite  number,  we 
may  conclude  that. 

When  the  multiplier  is  a  composite  number^  we  may  multi- 
ply by  each  of  the  factors  in  succession^  and  the  last  product 
will  he  the  entire  product  sought, 

•  Quest. — 61.  What  is  a  composite  number  ?  What  are  the  separate  parts 
called?  What  are  the  components  or  factors  in  the  number  12?  In  16? 
In  20  ?     How  do  you  proceed  when  the  multiplier  is  a  composite  number  ? 


72 


MULTIPLICATION. 


OPERATION. 
236 
4 
24  units. 
12    tens. 
8       hundreds. 

944 


62.  Let  it  be  required  to  multiply  236  by  4 ;  that  is,  to 
take  6  units,  3  tens,  and  2  hundreds,  each  4  times. 

First  set  down  the  236,  then  place  the 
4  under  the  unit's  place  6,  and  draw  a 
line  beneath  it.  Then  multiply  the  6 
units  by  4  :  the  product  is  24  units  ;  set 
them  down.  Next  multiply  the  3  tens  by 
4  :  the  product  is  12  tens  ;  set  down  the 
2  under  the  tens  of  the  24,  leaving  the  1 
to  the  left,  which  is  the  place  of  the  hun- 
dreds. Next  multiply  the  2  by  4 :  the  product  is  8,  which 
being  hundreds,  is  set  down  under  the  1.  The  sum  of  these 
numbers,  944,  is  the  entire  product. 

The  product  can  also  be  found,  thus : 
say  4  times  6  are  24 ;  set  down  the  4,  and 
then  say,  4  times  3  are  12  and  2  to  carry 
are  14 ;  set  down  the  4,  and  then  say,  4 
times  2  are  8  and  1  to  carry  are  9.  Set 
down  the  9,  and  the  product  is  944  as  before. 


OPERATION. 

236 

4 

944 


63.  Let  it  be  required  to  multiply  627  by  84. 

Multiply  by  the  4  units,  as  in  the  last 
example.  Then  multiply  by  the  8  tens. 
The  first  product  56,  is  56  tens  ;  the  6, 
therefore,  must  be  set  down  under  the  0, 
which  is  the  place  of  tens,  and  the  5  car- 
ried to  the  product  of  the  2  by  8.  Then 
multiply  the  6  by  8,  carry  the  2  from  the 
last  product,  and  set  down  the  result  50.  The  sum  of  the 
numbers,  52668,  is  the  required  product. 


OPERATION. 

627 

84 

2508 
5016 
52668 


64.  Let  it  be  required  to  multiply  £3  Ss.  6d.  Sfar.  by  6, 
in  which  each  of  the  denominate  numbers  is  to  be  taken  6 
times. 


Quest. — 62.  Explain  the  manner  of  multiplying  236  by  4.    63.  Explain 
thd  manner  of  multiplying  627  by  84. 


MULTIPLICATION.  73 


£    5.    d,  far. 

3     8     6     3 

6 

20  11     4     2 


We  first  say,  6  times  3  are  18  ;  that 
is,  18  farthings,  which  by  dividing  by  4 
are  found  equal  to  4c?.,  and  2  farthings 
over.  Set  down  the  2  farthings,  and 
then  say,  6  times  6  are  36,  and  4  to 
carry  make  40  ;  that  is,  40  pence,  which  after  dividing  by 
12,  are  found  equal  to  3  shillings  and  4  pence.  Set  down 
the  4d.,  and  then  say,  6  times  8  are  48  and  3  are  51  ;  that 
is,  51  shillings,  which  are  equal  to  £2  and  11  shillings  over. 
Set  down  the  11  shillings,  and  say,  6  times  3  are  18,  and  2 
to  carry  make  20,  which  write  under  the  pounds. 

65.  Hence,  to  multiply  one  number  by  another, 

Multiply  every  order  of  units  in  the  multiplicand,  in  success 
sion,  beginning  with  the  lowest,  by  each  figure  in  the  multi- 
plier, and  divide  each  product  so  formed  by  so  many  as  make 
one  unit  of  the  next  higher  denomination :  write  down  each  re- 
mainder under  units  of  its  own  order,  and  carry  the  quotient  to 
the  next  product, 

PROOF    OF    MULTIPLICATION. 

66.  Write  the  multiplier  in  the  place  of  the  multiplicand, 
and  find  the  product  as  before  ;  if  the  two  products  agree,  the 
work  may  be  supposed  right ;  Or, 

Divide  the  product  by  one  of  the  factors,  and  the  quotient 
^ill  be  the  other  factor. 

EXAMPLES. 

(1.)  (2.)  (3.)  (4.) 

847046  9807602  570409  216987 

8  7  8  6 


Quest. — 64.  Explain  the  manner  of  multiplying  £3  8s.  M.  Sfar.  by  6 
fy^.  What  is  the  general  rule  for  multiplication  ?     66.  What  is  the  first 
proof  of  multiplication  ?     What  is  the  second  ? 
•    4 


74  MULTIPLICATION. 

(5.)  (5.) 

Multiply  471493475  471493475 

by      4395  4395 


2357467375  1885973900 

4243441275  1414480425 

1414480425  4243441275 

1885973900  2357467375 


2072213822625  2072213822625 


Note  1.  Although  we  generally  begin  the  multiplication  by  the 
figure  of  the  lowest  denomination,  yet  we  may  multiply  in  any  or- 
der, if  we  only  preserve  the  places  of  the  different  orders  of  units. 
In  the  example  to  the  right,  we  began  with  the  order  of  thousands. 

Note  2.  Although  either  factor  may  be  used  as  the  multiplier, 
(Art.  60,)  still  it  is  best  to  use  that  one  which  contains  the  fewest 
places  of  figures,  as  is  shown  in  the  last  example.  For,  if  we 
change  the  process  and  use  the  multiplicand  as  the  multiplier,  there 
will  be  nine  multiplications  instead  of  four. 

6.  Multiply  430714934  by  743.  *  Ans,  

7.  Multiply  37157437  by  14972.  Ans,  

8.  Multiply  47157149  by  37049.  Arts.  

9.  Multiply  57104937  by  40709.  Ans.  — — 

10.  Multiply  79861207  by  890416.  Ans. 

11.  Multiply  9084076  by  9908807.  Ans.  


12.  Multiply  2748  by  200.  Ans,  549600. 

When  there  are  naughts  on  the  right  hand  of  the  signifi- 
cant figures  of  the  multiplier  or  multiplicand,  we  may  at  first 
neglect  them  in  the  multiplication ;  but  then  the  first  signifi- 
cant figure  of  the  product  will  be  of  a  higher  order  than  the 
first,  and  all  the  ciphers  must  be  added  in  order  to  reduce 
the  product  to  units  of  the  first  order. 

13.  Multiply  67046  by  10  :  also  by  100. 

14.  Multiply  57049  by  100  :  also  by  1000. 

15.  Multiply  4980496  by  1000:  also  by  10000 

16.  Multiply  90720400  by  100  :  also  by  10000 

17.  Multiply  74040900  by  1  :  also  by  10 

18.  Multiply  674936  by  100  :  also  by  100000. 


MULTIPLICATION.  75 

19.  Multiply  478400  by  270400.  Ans.  

20.  Multiply  367000  by  37409000.  Ans.  

21.  Multiply  7849000  by  84694000.  Ans,  

22.  Multiply  89999000  by  97770400.  Ans.  

23.  Multiply  9187416300  by  274987650000.  

24.  Multiply  86543291213456  by  12637482965.       

25.  Multiply  76729835645873  by  217834569.  

26.  Multiply  92413627858476  by  90587963412.       

27.  Multiply  87956743982714  by  819254837609.     

28.  Multiply  23869572491872  by  4007865347912.  — — 

29.  Multiply  68  by  the  composite  number  72.  


In  this  example  we  multiply  in  succession  by  the  factors 
9  and  8. 

30.  Multiply  3657  by  the  factors  of  64. 

31.  Multiply  37046  by  the  factors  of  121. 

32.  Multiply  2187406  by  the  factors  of  144. 

67.  In  multiplying  Federal  money  care  must  be  taken  to 
point  off  as  many  places  for  cents  and  mills  as  there  are  in 
the  multiplicand 

1.  Multiply  14  dollars  16  cents  and  8  mills,  by  5,  6, 
and  7. 

$14,168  $14,168  $14,168 

5  6  7 


(2.)         (3.)  (4.) 

$870,46      $894,120         $2141,096 

9  14  =  7  X  2        36  =6  X  6 


5.  What  will  95  pounds  of  tea  cost,  at  $1,04  per  pound? 

6.  What  will  105  yards  of  cloth  cost,  at  $3,25  per  yard  ? 

7.  What  will  four  firkins  of  butter  cost,  each  containing  97 
pounds,  at  25^  cents  per  pound  ?  .         ' 

Quest. — 67  What  precaution  is  necessary  in  multiplying  Federal  money  1 


76  MULTIPLICATION. 

8.  What  will  five  casks  of  wine  cost,  each  containing  59 
gallons,  at  $2,756  per  gallon  ? 

9.  A  bale  of  goods  contains  106  pieces,  costing  $55  and 
37-J  cents  each :  what  is  the  cost  of  the  entire  bale  ? 

10.  What  is  the  value  of  695  hats,  at  $3,654  each? 

11.  What  will  be  the  cost  of  97046  oranges,  at  2^  cents 
each? 

12.  What  will  be  the  cost  of  6742  sheep,  at  $2^  each? 

13.  What  will  be  the  cost  of  59  barrels  of  apples,  at  $2| 
per  barrel  ? 

14.  What  will  be  the  cost  of  6741  barrels  of  com,  at 
$3,254  per  barrel  ? 

BILLS    OF    PARCELS. 

15.  New  York,  May  1st,  1846. 

Mr,  James  Spendthrift 

Bought  of  Benj,  Saveall 

18  pounds  of  tea  at  85  cents  per  pound  -  -  -  - 
35  pounds  of  coffee  at  15^  cents  per  pound  -  -  - 
27  yards  of  linen  at  66  cents  per  yard       -     -     -     -^ 


Rec'd  payment,  Benj.  Saveall 


16.  Albany,  June  2d,  1846. 

Mr.  Jacob  Johns  Bought  of  Gideon  Gould, 

48  pounds  of  sugar  at  9^  cents  per  pound    -     -     -     - 
6  hogsheads  of  molasses,  63  gals,  each,  ?      ,     .     _ 
at  27  cents  a  gallon        -     -     .     -      ^ 

8  casks  of  rice,  285  pounds  each,  at  5  cts.  per  pound 

9  chests  of  tea,  86  pounds  each,  at  96  cts.  per  pound 


Total  cost 


"Rec'd  payment,  For  Gideon  Gould, 

Charles  Clark, 


MULTIPLICATION.  77 

17.  Hartford,  November  21st,  1846. 

Gideon  Jones  Bought  of  Jacob  Thrifty, 

78  chests  of  tea,  at  ^bbfib  per  chest 

251  bags  of  coffee,  100  pounds  each,  at  i 

12i  cts.  per  pound     -     -     -     - 
317  boxes  of  raisins,  at  $2,75  per  box       -     -     -     - 
1049  barrels  of  shad,  at  #7,50  per  barrel     -     -     -     - 
76  barrels  of  oil,  32  gallons  each,  at  $1,08  per  gal. 


Amount 

Received  the  above  in  full, 

Jacob  Thrifty, 

(18.) 

.  (19.) 

(20.) 

£    s,   d,        r. 

qr.   lb.     oz. 

yds.  ft.  in. 

Multiply    20     6     8i            3 

3     27     15 

16     2     9 

by                  4 

8 

9 

21.  What  will  4  yards  of 

25.  Soz.  at  75 

.  lOd. 

cloth   cost   at   7^.   6^dr  per 

26.  Sib.  at  7^ 

5W. 

yard? 

27.  10  gallons  at  16^.  4^df. 

22.  5  bushels  at  5^.  lOc^. 

28.   Wcwt.  at 

£1  9^.  lOirf. 

23.  6  yards  at  6s.  9d. 

29.   12  sheep 

at  £1  175.  9d, 

24.  7  ells  at  5^.  ll^d. 

30.  In  9  pieces  of  kersey,  each  \Ayds.  3qrs.  2na.,  how 
many  yards  ? 

31.  What  is  the  weight  of  12  tankards,  each  weighing 
II oz.  lOpwt.  I9gr.  ? 

32.  In  11  pieces  of  cloth,  each  \lyds.  Sqrs.  3wa.,  how  many 
yards  ? 

68.  In  multiplying  denominate  numbers,  if  the  multiplier 
is  a  composite  number,  and  greater  than  12,  it  is  best  to  mul- 
tiply by  the  factors  in  succession. 


Quest. — 68.  If  the  multiplier  is  a  composite  number,  how  should  you 
multiply  m  denominate  numbers  ? 


78 


MULTIPLICATION. 


37.  36 T.  at.  £5  15s.  Hid, 

38.  84  chaldrons  at£l  16;?. 

39.  108  bushels  at  7^.  9^d. 

40.  132  ells  at  18.9.  91J. 

41.  144  butts  at  £5  135.  Qi^;. 


33.  What  will  15  gallons 
of  wine  cost  at  5s.  3^d.  per 
gallon  ? 

34.  IShhds.  at  £3  I4s.  5d. 

35.  2iyds.  at  7^.  5^d. 

36.  35cwt.  at  £1  I7s.  8^d, 

42.  In  32  wedges  of  gold,  each  2llb,  7oz,  lAgr.,  how 
many  pounds  ? 

43.  In  21  fields,  each  3^.  2R,  19P.,  how  many  acres  ? 

69.  When  the  multiplier  is  greater  than  12  and  is  not  a 
composite  number. 

Take  the  nearest  composite  number  to  the  given  multiplier, 
and  multiply  by  its  factors  in  succession.  Then  multiply  by 
the  difference,  and  add  the  product  when  the  composite  num- 
ber is  less  than  the  multiplier,  and  subtract  it  when  greater. 

44.  What  is  the  cost  of  23  yards  of  cloth,  at  14^.  9d.  per 
yard? 

OPERATION. 

s.   d. 


s,   d, 

(14  9)x(7x3)+2 
7 

5     3  3  price  oil  yds. 
3 


Add 


15 
1 


9  9  price  of  21. 
9  6  price  of    2. 


Ans.£\%  19  3  price  of  23. 


Or  this,  (14  9)x(6x4)--l 
6 

4     8  6  price  of  6. 

4 


17  14  0  price  of  24. 
Subtract  14  9  price  of  1. 
Arts.  £16   19  3  23 


45.  What  is  the  cost  of  31  yards  at  12^.  lf\d.1 

46.  39  dozen  of  handkerchiefs  at  16^.  9\d, 

47.  139  pairs  of  stockings  at  4^.  9\d, 

48.  Sm.  of  silk  at  19^.  A.d. 

49.  Ill  sacks  of  flour  at  £1  As.  9^. 

50.  \5Qcwt.  at  £4  9^.  Qd.     v 


Quest. — 69.  How  do  you  multiply  when  the  multiplier  is  ^eater  than 
1 2  and  not  a  composite  number  ? 


MULTIPLICATION. 


79 


51.  In  57  years,  each  13  months,  1   day,  6  hours,  how 
many  months  1 

52.  What  is  the  weight  of  29hhds.  of  sugar,  each  weigh- 
ing 7cwt.2qr.  ISlb.l 

53.  In  67  parcels  of  tea,  each  25lb.  loz.  \Mr.,  how  many 
cwt.,  &c.  ? 

54.  What  will  394  yards  cost  at  \ls.  b\d,  per  yard? 

55.  357  calves  at  £7  105.  7c?. 
b^,  549  yards  at  12.9.  9\d, 
57.  754Z^.  of  tea  at  ^s.  10^. 
198ZZ>.  of  indigo  at  6^. 


OPERATION. 

s,     d. 
,17     5i 
10 


X  8 

14 

7  price  of  lOyds 
10 

87 

5 

10  price  of  100. 
3 

261 

78 

3 

6343 

17 

11 

9 

18 

6  price  of  300. 

3  price  of  90. 
10  price  of  4. 
"~7  price  of  394. 

58. 

59. 

\0d, 

60. 

61. 


754  weys    at   £20   5^. 


178  ells  at  bs,9\d. 
I98bbls.  at£l  14^.  9d. 
62.  744  chaldrons  at  £l  18^. 
Sd. 


70.  When  the  multiplier  has  a  fraction  annexed  to  it,  mul- 
tiply first  by  the  whole  number,  and  then  add  such  a  part  of 
the  multiplicand  as  the  fraction  is  of  unity. 


63.  What  will    56^    chal- 
•  drons  cost  at  £1  14^.  9d.  per 
chaldron  ? 

£   s.  d. 

1   14  9 

_7 

12     3  3    price  of  7. 


97     6  0   price  of  56. 
17  41-  price  of  i. 
Ans.  £98     34i  price  of  56i. 


64.  What  will  be  the  cost  of 
4f  yards  at  7^.  Qd.  per  yard? 

s,  d. 

7  6 

4 


1   10  0  price  of  4. 
4  2  price  of  |-. 


1   14  2  price  of  4|. 


9)37    6 
4    2 


Quest. — 70.  How  do  you  multiply  when  the  multiplier  has  a  fraction 
annexed  1 


80  MULTIPLICATION. 

65.  1788i  gallons  at  6^.  4d.  Ans. 

66.  SlUlcwi.  at  £4  ll^.  9d.  Ans. 

67.  7149|-  chaldrons  at  £1  14^.  9d.  Ans. 

68.  547f  lasts  at  £5  5s.  Ans. 
,  69.  1749i  firkins  at  14^.  d^d.  Ans. 
'      70.  754^cwt.  at  17^.  b^d.  Ans. 


BILLS    OF    PARCELS.  " 

71.  New  Orleans,  Jan.  2d,  1846. 

James  Lamh,  Esq. 

Bought  of  John  Simpson. 

.  £  s.  i 
7\  lbs.  of  green  tea  at  lOs.  Ad.  per  Ih.  -  -  -  - 
14i  do.  finest  bloom  at  14^.  Bd.  per  lb.  -  -  - 
10|-  do.  fine  green  at  16^.  5d.  per  lb.  -  -  - 
21  do.  hyson  at  10^.  I0\d.  per  lb.  -  -  -  - 
19  do.  good  hyson  at  13^.  9^d.  per  lb.  -  -  - 
8;!-     do.     bohea  at  6^.  9d.  per  lb. 


72.  Louisville,  March  19th,  1846. 

George  Veres,  Esq, 

Bought  of  Charles  West. 

£     s. 

A  loin  of  lamb,  weight  7^  lb.,  at  lOfc?.  per  lb.  -  - 

A  fillet  of  veal,  weight  16f  lb.,  at  6^d.  per  lb.   -  - 

A  buttock  of  beef,  weight  37^  lb.,  at  4^d.  per  lb.  - 

A  pig,  weight  12f  lb.,  at  7^d.  per  lb.        -     -     -  - 

A  leg  of  pork,  weight  16^  lb.,  at  5^d.  per  lb.     -  - 
A  leg  of  mutton,  weight  ISJ  lb.,  at  4^d.  per  lb. 


DIVISION,  81 


DIVISION. 

71.  Division  is  the  process  of  finding  how  many  times 
one  number  called  the  dividend  is  greater  or  less  than  another 
number  called  the  divisor ;  and  the  number  which  expresses 
how  many  times  the  dividend  is  greater  or  less  than  the  divi- 
sor, is  called  the;  quotient.  Hence,  the  quotient  is  as  many 
times  greater  or  less  than  unity,  as  the  dividend  is  greater  or 
less  than  the  divisor. 

72.  When  the  entire  quotient  can  be  expressed  by  a  whole 
number,  the  dividend  is  said  to  contain  the  divisor  an  exact 
number  of  times ;  but  when  it  cannot  be  so  expressed,  the 
part  of  the  dividend  which  remains  undivided  is  called  the 
remainder. 

73.  Since  the  quotient  shows  how  many  times  the  divi- 
dend exceeds  the  divisor,  it  follows,  that  if  ^the  divisor  be 
taken  as  many  times  as  there  are  units  in  the  quotient,  the 
product  will  be  equal  to  the  dividend.  And  hence,  if  the 
divisor  and  quotient  be  multiplied  together,  and  the  remain- 
der, if  any,  added  to  the  product,  the  result  will  be  equal  to 
the  dividend. 

EXAMPLES. 

1.  Divide  86  by  2. 

Place  the  divisor  on  the  left  of  the  divi- 
dend, draw  a  curved  line  between  them, 
and  a  straight  line  under  the  dividend. 

Now,  there  are  8  tens  and  6  units  to  be 
divided  by  2.  We  say,  2  in  8,  4  times, 
which  being  4  tens  we  write  the  4  under 


OPERATION. 

i  I 

eg     TS 

2)86 

43  quotient 


Quest. — 71,  What  is  division  ?  What  is  the  quotient?  How  many  times 
\b  it  greater  or  less  than  unity  ?  72.  When  can  the  entire  quotient  be  ex- 
pressed by  a  whole  number  ?  When  it  cannot,  what  do  you  call  the  part 
of  the  dividend  which  is  over?  73.  If  the  divisor  and  quotient  be  multiplied 
together,  what  will  the  product  be  equal  to  ? 

4* 


82 


DIVISION. 


OPERATION. 

3)7!^9 
243 


the  tens.  We  then  say,  2  in  6,  3  times,  which  are  three 
units,  and  must  be  written  under  the  6.  The  quotient,  there- 
fore, is  4  tens  and  3  units,  or  43.  Remark  that  each  order 
of  units  in  the  dividend,  on  being  divided,  gives  the  same  or- 
der of  units  in  the  quotient. 

2.  Divide  729  by  3. 

In  this  example  there  are  7  hundreds, 
2  tens,  and  9  units,  all  to  be  divided  by  3. 
Now,  we  say,  3  in  7,  2  times ;  that  is,  2 
hundreds,  and  1  hundred  over.  Set  down 
the  2  hundreds  under  the  7.  Now  of  the  7  hundreds  there 
is  1  hundred  or  10  tens  not  yet  divided.  We  put  the  10  tens 
with  the  2  tens,  making  it  12  tens,  and  then  say,  3  in  12,  4 
times  ;  that  is,  4  tens  times  ;  therefore  write  the  4  in  the  quo- 
tient, in  the  ten's  place ;  then  say,  3  in  9,  3  times.  The 
quotient,  therefore,  is  243. 

3.  Divide  729  by  9. 

In  this  example  we  say,  9  in  7  we  can- 
not, but  9  in  7^,  8  times,  which  are  8 
tens :  then,  9  in  9,  1  time. 

The  quotient  is  therefore  81. 

4.  Divide  8040  by  8. 
In  this  example  we  say,  8  in  8,  1  time, 

and  set  1  in  the  quotient.  We  then  say, 
8  in  0,  0  times,  and  set  the  0  in  the  quo- 
tient :  then  say,  8  in  4,  0  times,  and  set 
the  0  in  the  quotient :  then  say,  8  in  40,  5  times ;  that  is,  5 
units  times,  and  therefore  we  set  the  5  in  the  unit's  place  of 
the  quotient.     Therefore  the  true  quotient  is  1005. 

5.  Let  it  be  required  to  divide  36458  by  5. 

In  this  example,  we  find  the  quo- 
tient to  be  7291  and  a  remainder  3. 
This  3  ought  in  fact  to  be  divided 
by  the  divisor  5,  but  the  division 
cannot  be  effected,  since  3  does  not  contain  5.  The  division 
must  then  be  indicated  by  placing  5  under  the  3,  thus,  3. 


OPERATION. 

9)729 
8l 


OPERATION. 

8)8040 


1005 


OPERATION. 

5)36458 

7291—3  remain 


DIVISION. 


83 


OPERATION. 


n3  *j 


26)2756(106 
26  . 


The  entire  quotient,  therefore,  is  7291 1,  which  is  read,  seven 
thousand  two  hundred  and  ninety-one,  and  three  divided  by 
five.     Therefore, 

Where  there  is  a  remainder  after  the  division,- it  may  be 
written  after  the  quotient,  and  the  divisor  placed  under  it. 

74.  When  the  divisor  is  12  or  less  than  12,  the  operation 
may  be  performed  as  in  the  last  examples,  and  this  method 
of  dividing  is  called  short  division. 

6.  Divide  2756  by  26. 
We  first  say,  26  in  27  hundreds, 

once,  and  set  down  1  in  the  quotient, 
in  the  hundred's  place.  Multiplying 
by  1,  subtracting,  and  bringing  down 
the  5,  we  say,  26  in  15  tens,  0  tens 
times,  and  place  the  0  in  the  quo- 
tient. Bringing  down  the  6,  we 
find  that  the  divisor  is  contained 
in  156,  6  times.     Hence,  the  entire  quotient  is  106. 

7.  Divide  11772  by  327. 
Having  set  down  the  divisor  on 

the  left  of  the  dividend,  it  is  seen 

that  327  is  not  contained  in  the  first 

three  figures  on  the  left,  which  are 

117  hundreds.     But  by  observing 

that  3  is  contained  in  11,  3  times 

and  something  over,  we  conclude 

that-the  divisor  is  contained  at  least  3  times  in  the  first  four 

figures  of  the  dividend,  which  are  1177  tens.     Set  down  the 

3,  which  are  tens,  in  the  quotient,  and  multiply  the  divisor 

by  it:   we  thus  get  981  tens,  which  being  less  than  1177,  the 

quotient  figure  is  not  too  great :   we  subtract  the   981   tens 

from  the  first  four  figures  of  the  dividend,  and  find  a  remainder 

196  tens,  which  being  less  than  the  divisor,  the  quotient  fig- 


156 
156 


OPERATION. 

327)11772(36 
981 

1962 
1962 
0000 


Quest. — 74.  When  the  divisor  is  12  or  less  than  12,  what  is  the  division 
called? 


84  DIVISION. 

ure  is  not  too  small.     Reduce  this  remainder  to  units  and 
add  in  the  2,  and  we  have  1962. 

As  3  is  contained  in  19,  6  times,  we  conclude  that  the 
divisor  is  contained  in  1962  as  many  as  6  times.  Setting 
down  6  in  the  quotient  and  multiplying  the  divisor  by  it,  we 
find  the  product  to  be  1962.  Therefore  the  entire  quotient 
is  36,  or  the  divisor  is  contained  36  times  in  the  dividend. 

8.  Divide  £133  9^.  8d.  by  4. 

Here  we  again  take  the  least  num- 
ber of  units  of  the  highest  order  which 
will  contain  the  divisor,  viz.,  13  tens  of 
the  denomination  of  tens  of  pounds. 
Dividing  by  4,  we  find  the  quotient  to  be  3  tens  of  the  same 
denomination,  and  1  ten  over.  We  reduce  these  tens  to  units 
and  add  in  the  3,  and  thus  obtain  13  pounds,  which  being  di- 
vided by  4 'gives  3  pounds  and  1  over.  Reducing  this  £1  to 
shillings  and  adding  in  the  9,  gives  29,  which  being  divided  by 
4  gives  7  shillings  and  1  over.  Reducing  this  to  pence  and 
adding  in  8c?.,  and  again  dividing  by  4,  we  have  £33  7^.  5d 
for  the  entire  quotient. 


OPERATION. 

4)£133  9s.  8d. 
£33  7s.  5d. 


8)£6  8^.  8d. 
I6s.   Id. 


9.  Divide  £6  8s.  8d.  by  8. 
Here  we  have  to  pass  to  shillings  be- 
fore making  the  first  division. 

75.  Combining  the  principles  illustrated  in  the  foregoing 
examples  we  have,  for  the  division  of  numbers,  the  following  : 

Beginning  with  the  highest  order  of  units  of  the  dividend, 
pass  on  to  the  lower  orders  until  the  fewest  number  of  figures 
he  found  that  will  contain  the  divisor  :  divide  these  figures  hy 
it  for  the  first  figure  of  the  quotient :  the  unit  of  this  figure  will 
he  the  same  as  that  of  the  lowest  order  used  in  the  dividend. 

Multiply  the  divisor  hy  the  quotient  figure  so  found^  and 
subtract  the  product  from  the  dividend^  observing  to  place  units 
of  the  same  order  in  the  same  column.  Reduce  the  remainder 
to  units  of  the  next  lower  order ^  and  add  in  the  units  of  that  order 

Quest. — 75.  What  is  the  rule  for  the  division  of  numbers  ? 


DIVISION.  85 

found  in  the  dividend :  this  will  furnish  a  new  dividend.  Pro- 
ceed in  a  similar  manner  until  units  of  every  order  shall  have 
been  divided. 

76.  There  are  always  three  numbers  in  every  operation  of 
division,  and  sometimes  four.  First,  the  dividend ;  second, 
the  divisor ;  third,  the  quotient ;  and  fourth,  the  remainder, 
when  the  numbers  are  not  exactly  divisible. 

77.  There  are  five  operations  in  division.  First,  to  wiite 
down  the  numbers ;  second,  find  how  many  times ;  third,  mul- 
tiply ;  fourth,  subtract ;  and  fifth,  reduce  to  the  next  lower  order. 

EXAMPLES. 

1.  Divide  1203033  by  3679. 

By  the  first  operation,  300  times 
the  divisor  is  taken  from  the  divi- 
dend ;  or,  what  is  the  same  thing, 
the  divisor  is  taken  from  the  divi- 
dend 300  times.  By  the  second,  it 
is  taken  2  tens  or  twenty  times ; 
and  by  the  third,  it  is  taken  7  units 
times  ;  therefore,  it  is  taken  in  all  327  times  :  hence, 

78.  Divisioi^  is  a  short  method  of  performing  subtraction  ; 
and  the  quotient  found  according  to  the  rules  always  shows 
how  many  times  the  divisor  may  be  subtracted  from  the 
dividend. 

Prove  the  above  work  by  multiplying  the  divisor  and 
quotient  together. 

2.  Divide  714394756  by  1754.  Ans. 

3.  Divide  47159407184  by  3574.  Ans.  

4.  Divide  5719487194715  by  45705.  Ans.  

5.  Divide  4715714937149387  by  17493.  Ans.  

6.  Divide  671493471549375  by  47143.  Ans.  

7.  Divide  571943007145  by  37149.  Ans.  

Quest. — 76.  How  many  numbers  are  considered  in  division  ?  What  are 
they?  77.  How  many  operations  are  there  in  division?  Name  them. 
78.  How  may  division  be  defined  ;  and  wliat  does  the  quotient  show  ? 


OPERATION. 

3679)1203033(327 
11037 
9933 
7358' 

25753 
25753 


B6  DIVISION. 

8.  Divide  1714347149347  by  57143.      Ans,   

9.  Divide  49371547149375  by  374567.     Ans,   

10.  Divide  171493715947143  by  571007.    Ans.   

11.  Divide  6754371495671594  by  678957. 

12.  Divide  7149371478  by  121. 

13.  Divide  71900715708  by  57149. 

14.  Divide  15241578750190521  by  123456789. 

15.  Divide  121932631112635269  by  987654321. 

16.  Divide  14714937148475  by  123456. 

17.  Divide  8890896691492249389482962974  by  987675. 

PROOF  OF  MULTIPLICATION. 

79.  When  two  numbers  are  multiplied  together,  the  multi 
plicand  and  multiplier  are  both  factors  of  the  product ;  and 
if  the  product  be  divided  by  one  of  the  factors,  the  quotient 
will  be  the  other  factor.  Hence,  if  the  product  of  two  nuni' 
hers  be  divided  by  the  multiplicand,  the  quotient  will  be  the  mul- 
tiplier;  or,  if  it  be  divided  by  the  multiplier,  the  quotient  will 
he  the  multiplicand. 

1.  The  multiplicand  is  61835720,  the  product  8162315040  : 
what  is  the  multiplier  ? 

2.  The  multiplier  is  270000,  the  product  1315170000000  • 
what  is  the  multiplicand  ?  Ans.  

3.  The  product  is  68959488,  the  multiplier  96  :  what  is 
the  multiplicand? 

4.  The  multiplier  is  1440,  the  product  10264849920: 
what  is  the  multiplicand  ?  Ans.  

5.  The  product  is  6242102428164,  the  multiplicand 
6795634  :  what  is  the  multiplier  ?  Ans.  

80.  When  the  divisor  is  a  composite  number. 

Divide  the  dividend  by  one  of  the  factors  of  the  divisor,  and 
then  divide  the  quotient  thus  arising  by  the  other  factor :  the 
last  quotient  will  be  the  one  sought. 

Quest. — 79.  How  may  multiplication  be  proved  by  division  ?  80.  How 
do  you  divide  when  the  divisor  is  a  composite  number? 


DIVISION.  87 

EXAMPLES. 

1.  Let  it  be  required  to  divide  1407  dollars  equally  among 
21  men.     Here  the  factors  of  the  divisor  are  7  and  3. 

Let  the  1407  dollars  be  first  divi- 
ded equally  among  7  men.  Each 
share  will  be  201  dollars.  Let  each 
one  of  the  7  men  divide  his  share 
into  3  equal  parts,  each  one  of  the 
three  equal  parts  will  be  67  dollars,  and  the  whole  number 
of  parts  will  be  21  ;  here  the  true  quotient  is  found  by  divi- 
ding continually  by  the  factors. 

2.  Divide  18576  by  48  r=r  4  X  12.  Ans.  

3.  Divide  9576  by  72  zz:  9  x  8.  Ans.  


OPERATION. 
7)1407 
3)201   1st  quotient. 
67  quotient  sought 


4.  Divide  19290  by  96  =  12  X  8.  Ans.  

81.  It  sometimes  happens  that  there  are  remainders  after 
division — they  are  to  be  treated  as  follows : 

The  first  remainder,  if  there  he  one,  forms  a  part  of  the  true 
remainder.     The  product  of  the  second  remainder,  if  there  he 
one,  hy  the  first  divisor,  forms  a  second  part.     Either  of  these 
parts,  when  the  other  does  not  exist,  forms  the  true  remainder, 
and  their  sum  is^he  true  remainder  when  they  hoth  exist  together^ 
and  similarly  when  there  are  more  than  two  remainders. 
1.  What  is  the  quotient  of  751  grapes,  divided  by  16  ? 
(  4)751 
4  X  4  =  16  )  4)T87  ...  3 

C       46  ...  3  X  4  =r  12 
3 

15  the  true  remainder. 

Ans.  46^. 

In  751  grapes  there  are  187  sets,  (say  bunches,)  with  4 

grapes  or  units  in  each  bunch,  and  3  units  over.     In  the  187 

bunches  there  are  46  piles,  4  bunches  in  a  pile,  and  3  bunches 

over.     But  there  are  4  grapes  in  each  bunch ;  therefore,  the 

Quest. — 81.  How  do  y^ou  dispose  of  the  remainders,  if  there  are  anyi 
after  division  1 


88  DIVISION. 

number  of  grapes  in  the  3  bunches  is  equal  to  4  x  3  =  12, 
to  which  ad  1  3,  the  grapes  of  the  first  remainder,  and  we 
have  the  entire  remainder  15. 

2.  Divide  4967  by  32. 

r  4)4967 
4  X  8  =:  32  ^  8)1241   ...  3,  1st  remainder. 

(        155  ...  1  X  4  +  3  =  7  the  true  remainder* 

Ans.  155^. 

3.  Divide  956789  by  7  X  8  =:  56. 

4.  Divide  4870029  by  8  x  9  =  72. 

5.  Divide  674201  by  10  x  11  =  110. 

6.  Divide  445767  by  12  x  12  =  144. 

7.  Divide  375197351937  by  349J272  =  12x11  X  9  X  7 
X  7  X  6. 

12)375197351937 

11)31266445994... 9       -       -       -       -  =  9 

9)2842404181  ...  3  -  - 12  X  3      -       -       -        =  36 

7)315822686. ..7 --12x11x7     -       -        =        924 

7)451 17526...  4  --12x11x9x4      -        =.     4752 

6)6445360".  ..  6  --12x11x9x7x6        =    49896 

Quotient  =  1074226  .  . .  4  -  -  12  x  11  X  9  x7  X  7  x 4=  232848 

Remainder  =288465 


8.  Divide  7349473857  by  27.  Ans,  — — 

9.  Divide  749347549  by  144.  Ans,  

10.  Divide  649305743  by  55.  Ans.  

11.  Divide  4730715405  by  121.  Ans.  

12.  Divide  3704099714  by  108.  Ans.  

13.  Divide  4710437154  by  132.  Ans.  

14.  Divide  1071540075  by  99.  Ans.  

15.  Divide  468248  by  3x4x2x5x6. 

16.  Divide  98765432101234567890  by  12  X  11  X  10  X  9 
X8x7x6x5x4x3x2. 


116Q 
1367 
1305 


DIVISION.  89 

82.  When  the  divisor  has  one  or  more  O's  at  the  right,  it 
may  be  regarded  ag  a  composite  number,  of  which  one  factor 
is  1  with  as  many  O's  on  the  right  as  there  are  O's  at  the  right 
of  the  divisor,  and  the  remaining  figures  express  the  other 
factor.  StriJ^  off  the  0'^  and  the  same  number  of  figures  from 
the  right  of  the  dividend — this  is  dividing  by  one  of  the  factors ; 
then  proceed  to  divide  by  the  other, 

I.  Divide  14715967899  by  145000. 

145000)14715967899(101489j6f/^  Quotient. 
145  

215 

145  Or  thus, 

"^9      145000)14715967899(101489^^^ 

580  215 

1296  ~709 

1296 

~1369 

62899  Rem. 
62899  Rem.        

Note.  In  the  second  operation  of  this  example,  the  products  of 
the  divisor  by  each  quotient  figure  are  subtracted  mentally,  and  the 
remainders  only  written  down.  Let  the  pupil  perform  many  exam- 
ples in  division  in  this  way. 

2.  Divide  571436490075  by  36500. 

3.  Divide  194718490700  by  73000. 

4.  Divide  795498347594  by  47150. 

5.  Divide  1495070807149  by  31500. 

6.  Divide  6714934714934  by  754000. 

7.  Divide  1071491471430715  by  754000. 

8.  Divide  14714937493714957  by  157900. 

9.  Divide  7149374947194715  by  1749000. 
10.  Divide  714947349  by  90. 

II.  Divide  1714937148  by  14400. 

12.  Divide  69616103498721931800  by  97,5005700. 

Quest — 82.  What  is  the  (irocess  when  the  divisor  has  O's  annexed  * 


90  DIVISION. 

13.  Divide  656458931996524171800  by  700489070. 

14.  Divide  7149437149547  by  3714900. 

EXAMPLES    IN    DENOMINATE    NUMBERS. 

1.  A  gentleman's  income  is  £1260  15.y.  5d.  a  year:  what 
is  that  per  day,  365  days  being  contained  in  one  year  1 


£        s. 
365)1260  15 
1095 

d. 
5 

£ 
{  3 

s.    d. 

9  1  =  Ans, 
10 

165 
20 

34 

10  10  X  6 
10 

365)3315(95. 
3285 
30 
12 
365)  365  (IJ. 

345 

1036 

207 

17 

8  4 
3 
5  0 
5  0 
5  5 

365 

1260 

15  5  Proof. 

0 

2  Divide  £47  19^.  4d.  by  3.  Ans.  

3.  Divide  £37  14^.  lOd,  by  24.  Ans.  

4.  Divide  £49  19^.  U^d.  by  66.  Ans.  

*     5.  Divide  £34  14^.  d^d.  by  149.  Ans.  

6.  Divide  £1774  19^.  10^^.  by  179.  Ans.  

7.  Divide  47yd.  3qr.  2na.  by  5.  Ans.  

8.  Divide  37 A.  3R.  UP.  by  9.  Ans.  

9.  Divide  714^6.  IO02;.  I2gr.  by  89.  Ans.  

10.  Divide  374cwt.  3qr.  IQlb.  by  48.  Ans.  

11.  Divide  374E.E.  2qr.  3na.  by  142.  Ans.  

12.  If  60  sheep  be  sold  for  £112  10.?.,  what  is  the  value 
of  1  sheep  ? 

13.  If  ll2lh.  of  cheese  cost  £2  \Ss.  Sd.^  what  is  that  pet 
pound  ? 

14.  If  17 cwt.  of  lead  cost  £15  5^.  7^d.,  what  costs  Icwt.  ? 

15.  Bought  7  yards  of  cloth  for  16s.  4d. ;  what  is  that  per 
yard? 


DIVISION.  91 

16.  If  63  oxen  cost  £2553  1^.  6d,,  what  costs  1  ox? 

17.  U66lb.  of  butter  cost  £5  I5s.  6d.,  what  costs  llb.l 

18.  If  5281b.  of  tobacco  cost  £23  13^.,  what  costs  llb.l 

19.  If  a  tun,  or  252  gallons,  of  wine  cost  £60,  what  costs 
1  gallon  1 

20.  A  prize  of  1000  guineas  is  to  be  divided  among  150 
sailors  ;  what  is  each  man's  share,  after  deducting  |-  part  for 
the  officers  ? 

21.  If  125  ingots  of  silver,  each  of  an  equal  weight,  weigh 
l3^7oz.  llpwt.  I4:gr.,  what  is  the  weight  of  1  ingot? 

22.  If  475cwt.  \qr.  lAlb.,  be  the  weight  of  27hhds.  of  to- 
bacco, what  is  the  weight  of  Ihhd.  ? 

23.  Bought  6  pieces  of  tapestry,  containing  237-B.  Fl.  2qr, 
2na. ;  what  is  the  length  of  1  piece  ? 

APPLICATIONS. 

1.  In  1842,  nine  mills  in  Lowell  manufactured  434000 
pounds  of  cotton  per  week.  How  much  was  manufactured 
by  each  mill,  supposing  the  amount  was  exactly  the  same  ? 

2.  The  number  of  inhabitants  in  the  city  of  New  York  in 
1840  was  312710,  and  the  expenses  of  the  city  government 
$1645779,30.  If  this  was  raised  by  an  equal  tax  upon  every 
inhabitant,  how  much  would  each  have  to  pay  ? 

3.  The  number  of  hogsheads  of  tobacco  exported  from  the 
United  States  in  the  20  years  preceding  1841,  was  1792000, 
and  their  estimated  value  was  $131346514.  What  was  the 
average  value  by  the  hogshead  ? 

4.  The  amount  of  coffee  imported  in  1840  was  94996095 
poimds,  and  its  value  was  estimated  at  $8546222.  What 
was  its  worth  per  pound  ? 

5.  The  number  of  scholars  attending  the  public  schools  of 
the  state  of  Maine,  in  1839,  was  201024,  and  the  amount  ex- 
pended for  the  support  of  the  schools  was  $2581 13,43.  What 
was  the  cos*  to  the  state  for  the  tuition  of  each  scholar  ? 


92  DIVISION. 

6.  The  militia  force  of  the  United  States,  according  to  the 
Army  Register  for  1845,  was  1426868,  and  the  number  of 
commissioned  officers  belonging  to  it  was  69450.  How  many 
soldiers  did  that  allow  to  each  officer  ? 

7.  The  whole  coinage  of  the  United  States  for  the  51  years 
preceding  1845,  amounted  to  $110177761,38.  Suppose  an 
equal  amount  had  been  coined  each  year,  what  would  it  have 
been? 

8.  In  1843  there  were  sold  1605264  acres  of  the  public 
lands.  The  sum  received  for  them  was  $2016044,30,  and 
the  sum  paid  into  the  national  treasury,  after  deducting 
expenses,  was  $1997351,57.  What  was  the  average  cost 
per  acre  to  the  purchasers,  and  what  was  the  average  price 
per  acre  received  by  the  government  ? 

9.  The  net  amount  of  duties  on  imports  for  eighteen  years 
preceding  1843,  was  $452539300,81.  How  much  was  col- 
lected in  each  year,  supposing  the  sums  to  have  been  equal  1 

10.  There  was  inspected  in  Onondaga  county,  N.  Y.,  in 
1844, 4003554  bushels  of  salt.  The  duties  collected  on  these 
amounted  to  $240305.     What  was  the  duty  on  each  bushel  ? 

11.  There  were  thirty-five  banks  in  New  Hampshire  in 
1844,  whose  whole  resources  were  $5836014,07.  If  this 
sum  was  equally  divided,  how  much  would  belong  to  each  ? 

12.  The  population  of  Europe  .in  1837  was  estimated  at 
233884800,  and  the  number  of  square  miles  at  3708871. 
How  many  inhabitants  would  this  give  to  each  square  mile  ? 

13.  In  1843,  there  were  3173  public  schools  in  Massa- 
chusetts, which  were  attended  during  the  winter  by  1 1 9989 

scholars.     How  many  would  this  allow  to  each  .school? 
# 

14.  The  number  of  male  scholars  attending  the  public 
schools  of  Pennsylvania  was  reported,  in  1843,  to  be  161164 
and  the  number  of  female  scholars  127598.  The  number  of 
male  teachers  employed  by  the  state  was  5264,  and  the  num- 
ber of  female  teachers  2330.  How  many  scholars  would 
this  give  to  each  teacher  ? 


•«» 


PROPERTIES    OF    THE    9's.  93 

15.  The  value  of  the  exports  from  the  United  States  in 
1841,  was  $104691534.  If  an  equal  amount  had  been  ex- 
ported each  day  of  the  year  excepting  Sundays/ what  would 
it  have  been  1 

OF    THE    PROPERTIES    OF    THE    9's. 

83.  Besides  the  methods  already  explained  of  proving  the 
operations  in  figures,  there  is  yet  another  called  the  method 
by  casting  out  the  9^s.     That  method  we  will  now  explain. 

84.  An  excess  of  units  over  exact  9's,  is  the  remainder 
after  the  number  has  been  divided  by  9  :  hence,  any  number 
less  than  9  must  be  treated  as  an  excess  over  exact  9's. 

Let  us  write  down  the  numbers 
to  be  added,  as  at  the  right.  Now, 
if  we  divide  each  number  by  9, 
and  place  the  quotients  to  the  right, 
and  the  remainders  in  the  column 
still  to  the  right,  we  shall  have,  in 
the  middle  column,  the  exact  num- 
ber of  9's  contained  in  each  num- 
ber, and  in  the  column  at  the  right, 
the  excesses  over  exact  9's.  By 
adding  these  columns,  we  find  15 
in  the  column  of  remainders,  which  is  equal  to  one  9  and  6 
over :  hence,  there  are  764  exact  9's  and  6  over.  But  it  is 
evident  that  the  sum  of  all  the  numbers,  viz.,  6882,  must  con- 
tain exactly  the  same  number  of  9's  and  the  same  excess 
over  exact  9's,  as  are  found  in  the  numbers  taken  separately, 
since  a  whole  is  equal  to  the  sum  of  all  its  parts  any  way 
taken  :  therefore,  in  the  sum  of  any  numbers  whatever,  the 
number  of  exact  9'^  and  the  excess  over  9'^  are  equal,  respect- 
ively, to  the  aggregate  of  exact  9^s' and  the  excess  of  9'^  in  the 
numbers  taken  separately. 


OPERATION. 

^ 

fe  . 

^O) 

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<^  >» 

03^ 

£  ^ 

la 

^S 

s  .a 

c  <u 

©-a 

'S'rl 

§         ^ 

s  s 

a 

S 

3870  ...  430  . 

.  .  0 

2708  ...  300  . 

.  .  8 

304  ..  .    33  . 

.  .  7 

)6882          764 

6 

764-6 

Quest. — 83.  What  other  methods  of  proof  are  there  for  arithmetical 
oper^ions,  besides  those  already  explained?  84.  What  is  an  excess  of  9's? 
How  do  the  exact  number  of  9's  and  the  excess  of  9's  in  any  sum  com- 
pare with  the  exact  9's  and  the  excess  of  9's  in  the  several  numbers? 


94  PROPERTIES    OF    THE    9's. 

85.  We  will  now  explain  a  short  process  of  finding  the 
excess  over  an  exact  number  of  9's  in  any  number  whatever ; 
and  to  do  this,  we  must  look  a  little  into  the  formation  of 
numbers. 

In  any  number,  written  wdth  .a  single  significant  figure,  aa 
4,  50,  600,  8000,  &c.,  the  excess  over  exact  9's  will  always  be 
equal  to  the  number  of  units  expressed  by  the  significant  figure ; 
for,  in  any  such  number  we  shall  always  have  4  =  4 

Also, 50  =  (9     +1)X5 

600  =  (99  +1)X6 

8000  =  (999+1)  X  8 
&c.  &c.  &c. 

Each  of  the  numbers  9,  99,  999,  &c.,  expresses  an  exact 
number  of  9's  ;  and  hence,  when  multiplied  by  5,-6,  8,  &c., . 
the  several  products  will  each  contain  an  exact  number  of 
9's ;  therefore,  the  excess  over  exact  9's,  in  each  number, 
will  be  expressed  by  4,-5,  6,  8,  &c. 

If,  then,  we  write  any  number  whatever,  as 
6253, 
we  may  read  it  6  thousand  2  hundred  50  and  3.  Now,  the 
excess  of  9's  in  the  6  thousand  is  6  ;  in  2  hundred  it  is  2 ; 
in  50  it  is  5  ;  and  in  3  it  is  3  :  hence,  in  them  all,  it  is  16,- 
which  makes  one  9  and  7  over :  therefore,  7  is  the  excess 
over  exact  9's  in  the  number  6253.  Hence,  the  excess  over 
exact  9'^  in  any  number  whatever,  may  he  found  by  adding  to- 
gether the  significant  figures,  and  rejecting  the  exact  9'^  from 
the  sum. 

Note. — It  is  best  to  reject  or  drop  the  9  as  soon  as  it  occurs  : 
thus  we  say,  3  and  5  are  8  and  2  are  10 ;  then  dropping  the  9,  we 
say,  1  to  6  is  7,  which  is  the  excess  ;  and  the  same  for  all  similar 
operations. 

1.  What  is  the  excess  of  9's  in  48701  ?     In  67498  \ 

2.  What  is  the  excess  of  9's  in  9472021  ?     In  2704962  1 

3.  What  is  the  excess  of  9's  in  87049612  ?     In  4987051  ? 

Quest. — 85.  What  will  be  the  excess  over  exact  9's  in  any  number  ex- 
pressed by  a  single  significant  figure  ?  How  may  the  excess  over  exact  9*s 
be  found  in  any  number  whatever  1 


PROPERTIES    OF    THE 


9's. 


95 


OPERATION. 

Excess  of  9's» 

94874  . 

.  5 

46073   . 

.  2 

50498  . 

.  8 

3674  . 

.  2 

341   . 

.  8 

195460-7 

7 

PROOF    OF    ADDITION    BY    CASTING    OUT    THE    9's. 

86. — 1.  In  the  first  of  these  num- 
bers we  find  the  excess  of  9's  to  be  5  ; 
in  the  second  2  ;  in  the  third  8.;  in 
the  fourth  2  ;  and  in  the  fifth  8 : 
hence,  in  them  ail  it  is  25,  which 
leaves  7  for  the  excess  over  exact 
9's.  We  also  find  7  to  be  the  excess 
over  exact  9's  in  the  sum  195460: 
hence  the  work  is  supposed  to  be  right.  Notwithstanding 
this  proof,  it  is  possible,  after  all,  that  the  work  may  be  erro- 
neous. For  example,  if  either  figure  in  the  sum  is  too  large 
by  one  or  more  units,  and  any  other  figure  is  too  small  by  the 
same  number  of  units,  the  excess  over  exact  9's  will  not  be 
affected.  But  as  it  would  seldom  happen  that  one  error 
would  be  exactly  balanced  by  another,  the  work  when  proved 
may  be  relied  on  as  correct.  Similar  sources  of  error  exist 
*  I  the  proof  of  all  arithmetical  operations. 

2.  Add  together,  8754608,  4908721,  6027983,  89704543, 
3142367,  and  28949760,  and  prove  the  result  by  rejecting 
Oie  9's. 

3.  Add  together  40799903,  874162,  32704931,  6704192, 
2146748,  94004169,  and  prove  the  result  by  casting  out 
the  9's. 


OPERATION. 

874136  ...  2 
45302   ...  5^ 

828834  .   .   .  6* 


PROOF    OF    SUBTRACTION    BY    CASTING    OUT    THE    9's. 

87. — 1.  Since  the  sums  of  the  re- 
mainder and  subtrahend  must  be  equal 
to  the  minuend,  it  follows  that  the  ex- 
cess of  9's  in  these  two  numbers  must 
be  equal  to  the  excess  of  9's  in*  the 
minuend :  hence,  to  the  excess  ofd^s  in  the  remainder  add  the 
excess  of  9'^  in  the  subtrahend,  and  the  excess  of  9'6'  m  the 
sum  will  he  equal  to  the  excess  of  9'^  in  the  minuend. 

Quest. — 86.  Explain  the  proof  of  addition  by  casting  out  the  9's.  In 
what  is  the  proof  defective?  87.  Explain  the  proof  of  subtraction  by  cast- 
ing out  the  9's. 


96  PROPERTIES    OF    THE    9  S. 

2.  From  874096  take  370494,  and  prove  the  work  by  re- 
jecting the  9's, 

3.  From  47096702  take  1104967,  and  prove  the  work  by 
rejecting  the  9's. 

PROOF    OF    MULTIPLICATION    BY    CASTING    OUT    THE    9's. 

88.  We  will  first  remark,  that  if  any  number  containing 
an  exact  number  of  9's  be  multiplied  by  another  whote 
number,  the  product  will  also  contain  an  exact  number 
of  9's. 

Let  it  be  required  to  multiply  any  two  numbers  together, 
as  641  and  232. 

We  first  find  the  excess  over  exact 
9's  in  both  factors,  and  then  separate 
each  factor  into  two  parts,  one  of 
which  shall  contain  exact  9's,  and 
the  other  the  excess,  and  unite  the 
two  together  by  the  sign  plus.  It  is 
now  required  to  take  639 +2==  641, 
IS  many  times  as  there  are  units  in 
225  +  7  r=  232. 

Beginning  with  the  7,  we  have  14 
for  the  product  of  2  by  7,  and  4473  for  the  product  of  639  by 
7  ;  and  this  last  contains  an  exact  number  of  9's.  We  then 
take  2,  225  times,  which  gives  450,  which  also  contains  an 
exact  number  of  9's.  We  next  multiply  639  by  the  figures 
of  225,  and  each  of  the  several  products  contains  an  exact 
number  of  9's,  since  639  contains  an  exact  number.  Hence, 
the  entire  sum  148698  contains  an  exact  number  of  9's,  to 
which  if  we  add  the  one  9.  from  the  14,  we  shall  find  the  ex- 
cess of  9's  in  the  product  to  be  5  ;  and  as  the  same  may  be 
shown  for  any  numbers,  we  conclude  that,  the  excess  of  9'^  in 
any  product  must  arise  from  the  product  of  the  excess  of9^s  in 
the  factors. 

Quest. — 88.  Explain  the  proof  of  multiplication  by  casting  out  the  9'». 
What  does  the  excess  of  9's  in  any  product  arise  from? 


OPERATION. 

641  =639  -\-    2 

232  :r=  225  +  7 

4473  -f  14 
450 

3195 

.  1278 

1278 

148698  +  14 

PROPERTIES    OF    THE    9's.  97 

But  since  the  product  of  two  numbers  found  in  the  ordinary- 
way  must  contain  the  same  number  of  9's,  and  the  same  ex- 
cess  of  9's  as  a  product  found  above,  it  follows  that,  if  the 
excesses  of  9'.y  in  any  number  of  factors  he  multiplied  together, 
the  excess  of  9'^  in  such  product  will  be  equal  to  the  excess  of 
9'*  in  the  product  of  the  factors . 

EXAMPLES. 

(1.)  (2.) 

Multiply           87603  ...  6  818327  ...  2 

by  9865  1  9874 1^ 

Prod.   864203595  ...  6  8080160798  ...  2 


3.  By  multiplication  we  have 

Ex.4.         Ex.8.       Ex.4.         Ex.  of  product,  2. 

7285  X  143  X  976  =  1016752880. 

Ex.5.      Ex.4.      Ex.0.  Ex.0. 

4.  We  also  have    869  x  49  x  36  =  1532916. 

When  the  excess  of  9's  in  any  factor  is  0,  the  excess  of 
9's  in  the  product  is  always  0. 

PROOF    OF    DIVISION    BY   CASTING   OUT    THE    9's. 

89.  Since  the  divisor  multiplied  by  the  quotient  must  pro- 
duce the  dividend,  it  follows  that  if  the  excess  of  9's  in  the 
divisor  be  multiplied  by  the  excess  of  the  9's  in  the  quotient, 
the  excess  of  9's  in  the  product  will  be  equal  to  the  excess 
of  9's  in  the  dividend. 

1.  The  dividend  is  8162315040,  the  divisor  61835720,  and 
the  quotient  132  ;  is  the  work  right? 

2.  The  dividend  is  10264849920,  the  divisor  1440,  and 
the  quotient  7128368  :  is  the  work  right  ? 

3.  The  dividend  is  74855092410,  the  quotient  78795,  and 
the  divisor  949998  :  is  the  work  right  ? 

Let  the  pupils  apply  the  property  of  the  9's  to  other  ex- 
amples. 

Quest. — If  the  excess  of  9's  in  any  number  of  factors  be  multiplied  to- 
gether, what  will  the  excess  of  9's  in  the  product  be  equal  to?  89.  How  do 
you  prove  division  by  casting  out  the  9's  ? 

5 


98  REMARKS. 

REMARKS. 

90. — 1 .  Numeration,  Addition,  Subtraction,  Multiplication, 
and  Division  are  called  the  five  ground  rules,  because  all  the 
other  operations  of  arithmetic  are  performed  by  means  of 
them.  Multiplication,  however,  is  but  a  short  method  of  per- 
forming addition,  and  division  but  an  abridged  method  of 
subtraction. 

2.  A  prime  number  is  one  which  cannot  be  exactly  divi- 
ded by  any  number  except  itself  and  unity.  Thus,  1,  3,  5, 
7,  11,  13,  19,  23,  &c.,  are  prime  numbers. 

3.  The  product  of  two  or  more  prime  numbers  will  be  ex- 
actly divisible  only  by  one  or  the  other  of  the  factors. 

4.  If  an  even  number  be  added  to  itself  any  number  of 
times,  the  sum  will  be  even ;  hence,  if  one  of  the  factors  of 
a  product  be  an  even  number,  the  product  will  be  even. 

5.  An  odd  number  is  not  divisible  by  an  even  number ;  nor 
is  a  less  number  exactly  divisible  by  a  greater. 

6.  The  quotient  arising  from  the  division  of  the  sum  of  two 
or  more  numbers,  by  any  divisor,  is  equal  to  the  sum  of  the 
quotients  which  arise  from  the  division  of  the  parts  separately. 

7.  Any  number  is  divisible  by  2,  if  the  last  significant 
figure  is  even ;  and  is  divisible  by  4,  if  the  last  two  figures  are 
divisible  by  4. 

8.  Any  number  whose  last  figure  is  5  or  0,  is  exactly 
divisible  by  5 ;  and  any  number  whose  last  figure  is  0,  is 
exactly  divisible  by  10. 

Quest. — 90. — 1.  What  are  the  five  ground  rules  of  arithmetic  ?  What 
other  rule  in  fact  embraces  the  rule  of  multiplication  ?  How  may  division 
be  performed  ?  2.  What  is  a  prime  number  ?  3.  By  what  numbers  only  will 
the  product  of  prime  factors  be  divisible  ?  4.  If  an  even  number  be  multi- 
plied by  a  whole  number,  will  the  product  be  odd  or  even  ?  5.  Is  an  odd 
number  divisible  by  an  even  number  ?  6.  What  is  the  quotient  arising  from 
the  division  of  the  sum  of  two  or  more  numbers  by  any  divisor  equal  to  ? 
7.  When  is  a  number  exactly  divisible  by  2  ?  When  by  4?  8.  If  the  last 
figure  of  a  number  be  5  or  0,  by  what  numbers  may  it  be  divided  ? 


mVIStONS    OP    AHITHMETIC.  99 

DIVISIONS   OF  ARITHMETIC. 

91.  The  science  of  arithmetic,  which  treats  of  numbers,  may 
be  divided  into  four  parts : 

1st.  That  which  treats  of  the  properties  of  entire  units,  called 
the  Arithmetic  of  Whole  Numbers ; 

2d.  That  which  treats  of  the  parts  of  unity,  called  the  Aiith- 
metic  of  Fractions ; 

3d.  That  which  treats  of  the  relations  of  the  unit  1  to  the 
numbers  which  come  from  it,  whether  they  be  integers  or 
fractions,  and  the  relations  of  these  numbers  to  each  other ;  and 

4th.  The  application  of  the  science  of  numbers  to  practical 
and  useful  purposes. 

A  portion  of  the  First  part  has  already  been  treated  under 
the  heads  of  Numeration,  Addition,  Subtraction,  Multiplication 
and  Division. 

The  Second  part  comes  next  in  order,  and  naturally  divides 
itself  into  two  branches;  viz., 

Vulgar  or  Common  Fractions,  in  which  the  unit  is  divided 
into  any  number  of  equal  parts,  and  Decimal  Fractions,  in 
which  the  unit  is  divided  according  to  the  scale  of  tens. 

The  Third  part  relates  to  the  comparison  of  numbers,  with 
respect  either  to  their  difference  or  quotient.  The  Rule  of 
Three,  and  Arithmetical  and  Geometrical  Progression,  make  up 
this  branch  of  Arithmetic. 

The  Fourth  part  embraces  the  applications  of  rules  deduced 
fi-om  the  science  of  numbers,  to  the  ordinary  transactions  and 
business  of  life. 

106.  Of  what  does  the  science  of  arithmetic  treat?  Into  how  many 
parts  may  it  be  divided  ?  Of  what  does  the  first  part  treat  ?  Of  what 
does  the  second  part  treat  ?  What  is  it  called  ?  Of  what  does  the  third 
part  treat  ?  What  does  the  fourth  part  embrace  ?  Which  part  has  been 
treated  ?  Under  how  many  heads  ?  Into  how  many  heads  is  the  second 
part  divided  ?  What  are  they  called  ?  What  distinguishes  them  ?  To 
what  does  the  third  part  relate  ?     What  does  the  fourth  part  embrace  ? 


100  OF   VULGAR    FRACTIONS. 

OF  VULGAR  FRACTIONS. 

92.  The  unit  1  represents  an  entire  thing,  as  I  apple, 
1  chair,  1  pound  of  tea. 

If  we  suppose  1  thing,  as  1  apple,  or  1  pound  of  tea,  to  be 
divided  into  two  equal  parts,  each  part  is  called  one  half  oi 
the  thing. 

If  the  unit  be  divided  into  3  equal  parts,  each  part  is  called 
one  third. 

If  the  unit  be  divided  into  4  equal  parts,  each  part  is  called 
one  fourth. 

If  the  unit  be  divided  into  12  equal  parts,  each  part  is 
called  one  twelfth ;  and  when  it  is  divided  into  any  number 
of  equal  parts,  we  have  a  similar  expression  for  each  of  the 
parts.     These  equal  parts  of  a  unit  are  called  Fractions. 

How  are  these  fractions  to  be  expressed  by  figures  ?  They 
are  expressed  by  writing  one  figure  under  another.     Thus, 

•J-   is  read  one  half.  |    \    is  read   one  seventh. 

•i    "     "      one  third.  I    -g-     "     "      one  eighth. 

\    "     **      one  fourth.  I   ts"    "     "      oi^®  tenth. 

\    "     "       one  fifth.  j   ^^    "     "      one  fifteenth. 

\    "     "      one  sixth.  \   it    ''     "      one  fiftieth. 

It  should,  however,  be  observed,  that  -^  is  an  entire  half; 
I",  an  entire  third,  and  the  same  for  all  the  other  fractions. 
Now,  these  fractions  being  entire  things^  may  be  regarded  as 
units,  and  each  is  called  a  fractional  unit. 

93.  It  is  thus  seen  that  every  fraction  is  expressed  by  two 
numbers.  The  number  which  is  written  above  the  line  is 
called  the  numerator,  and  the  one  below  it,  the  denominator, 
because  it  gives  a  denomination  or  name  to  the  fraction. 

For  example,  in  the  fraction  i,  1  is  the  numerator,  and  2 


Quest. — 92.  What  does  the  unit  1  represent  ?  If  we  divide  it  into  two 
equal  parts,  what  is  each  called?  If  it  be  divided  into  three  equal  parts, 
what  is  each  part  ?  Into  4,  5,  6,  &c.,  parts  ?  What  are  such  expressions 
called?  How  may  the  fractions  be  regarded?  What  are  they  called? 
93.  Of  how  many  numbers  is  each  fraction  made  up  ?  What  is  the  one 
above  the  line  called  ?     The  one  below  the  line  ? 


OF    VULGAR    FRACTIONS.  101 

the  denominator.     In  the  fraction  ^,  1  is  the  numerator,  and 
3  the  denominator. 

The  denominator  in  every  fraction  shows  into  how  many  equal 
parts  the  unit,  or  single  thing,  is  divided.  For  example,  in  the 
fraction  ^,  the  unit  is  divided  into  2  equal  parts  ;  in  the  frac- 
tion i,  it  is  divided  into  three  equal  parts  ;  in  the  fraction  \, 
it  is  divided  into  four  equal  parts,  &c.  In  each  of  the  above 
fractions  one  of  the  equal  parts  is  expressed. 

But  suppose  it  were  required  to  express  2  of  the  equal 
parts,  as  2  halves,  2  thirds,  2  fourths,  &;c. 
We  should  then  write, 

I     they  are  read     two  halves. 
I       "       "       "       two  thirds. 
I       "       "       "       two  fourths. 
I       "       "       "       two  fifths,  &c. 
If  it  were  required  to  express  three  of  the  equal  parts,  we 
should  place  3  in  the  numerator ;  and  generally,  the  numera- 
tor shoios  how  many  of  the  equal  parts  are  expressed  in  the 
fraction. 

For  example,  three  eighths  are  written, 
|-     and  read     three  eighths. 
|-       "      "        four  ninths, 
-j^     "      "         six  thirteenths. 
^      "      "         nine   twentieths. 

94.  When  the  numerator  and  denominator  are  equal,  the 
numerator  will  express  all  the  equal  parts  into  which  the  unit 
has  been  divided  :  and,  the  value  of  the  fraction  is  then  equal 
to  1.  But  if  we  suppose  a  second  unit, -of  the  same  kind,  to 
be  divided  into  the  same  number  of  equal  parts,  those  parts 


Quest. — What  does  the  denominator  show  ?  What  does  the  numerator 
show  ?  In  the  fraction  three-eighths,  which  is  the  numerator  ?  Which  the 
denominator  ?  Into  how  many  parts  is  tlie  unit  divided  ?  How  many 
parts  are  expressed  ?  In  the  fraction  nine-twentieths,  into  how  many  parts 
is  the  unit  divided  ?  How  many  parts  are  expressed  ?  94.  When  the 
numerator  and  denominator  are  equal,  what  is  the  value  of  the  fraction? 


102 


OF    VULGAR    FRACTIONS. 


may  also  be  expressed  in  the  same  fraction  with  the  parts  of 
the  first  unit.     Thus, 

f      is  read     three  halves. 

■J       "     "        seven  fourths. 

V^     "     "        sixteen  fifths. 

V^      "     "        eighteen  sixths. 

Y^      "     "        twenty-five  sevenths. 

If  the  numerator  of  a  fraction  be  divided  by  its  denominator, 
the  integer  part  of  the  quotient  will  express  the  number  of 
entire  units  which  have  been  used  in  forming  the  fraction,  and 
the  remainder  will  show,  how  many  fractional  units  are  over. 

The  unit,  or  whole  thing,  which  is  divided,  in  forming  a 
fraction,  is  called  the  unit  of  the  fraction ;  and  one  of  the  equal 
parts  is  called  the  ^mit  of  the  expression.  Thus,  in  the  fraction 
I,  1  is  the  unit  of  the  fraction,  and  -j-  the  unit  of  the  expression. 

In  every  fraction,  we  must  distinguish  carefully,  between 
the  unit  of  the  fraction  and  the  unit  of  the  expression.  The 
first,  is  the  whole  thing  from  which  the  fraction  is  derived ;  the 
second,  one  of  the  equal  parts  of  the  fractional  expression. 

From  what  has  been  said,  we  conclude  : 

1st.  That  a  fraction  is  the  expression  of  one  or  more  equal 
parts  of  unity. 

2d.  That  the  denominator  of  a  fraction  shows  into  how 
many  equal  parts  the  unit  or  single  thing  has  been  divided^ 
and  the  numerator  expresses  how  many  such  parts  are  taken 
in  the  fraction. 

3d.  That  the  value  of  every  fraction  is  equal  to  the  quotient 
arising  from  dividing   the  numerator  by  the  denominator, 

4th.  That^  when  the  numerator  is  less  than  the  denomina- 
tor, the  value  of  the  fraction  is  less  than  1. 

Quest. — What  is  the  value  of  the  fraction  three-halves?  Of  seven- 
fourths  ?  Of  sixteeu-fifths  ?  Of  eighteen-sixths  ?  Of  twenty-five-sevenths  ? 
What  is  the  first  conclusion  ?  What  the  2d  ?  What  the  3d  ?  What 
the  4th?  What  the  5th?  What  the  6th?  What  the  7th?  What  is 
the  unit  of  the  fraction  three-fourths  ?  What  is  the  unit  of  the  expres- 
uion? 


OF    VULGAR    FRACTIONS.  103 

5tli.  That^  when  the  numerator  is  equal  to  the  denominator^ 
the  value  of  the  fraction  is  equal  to  \. 

6th.  That,  when  the  numerator  is  greater  than  the  denomi- 
nator.  the  value  of  the  fraction  will  be  greater  than  1. 

Yth.  That,  the  unit  of  every  fraction  is  the  whole  thing 
from  which  it  was  derived;  and  the  unit  of  the  expression^ 
one  of  the  equal  parts  taken, 

95.  There  are  six  kinds  of  Vulgar  Fractions  :  Proper,  Im- 
proper, Simple,  Compound,  Mixed,  and  Complex. 

A  Proper  Fraction  is  one  in  which  the  numerator  is  less 
than  the  denominator.  The  value  of  every  proper  fraction  is 
Jess  than  1,  (Art.  94). 

The  following  are  proper  fractions  : 

111335  9  85 

2'      3»      4»      4'      7'      ¥»      10»      9»      6* 

An  Improper  Fraction  is  one  in  which  the  numerator  is 
equal  to,  or  exceeds  the  denominator.  Such  fractions  are 
called  improper  fractions  because  they  are  equal  to,  or  ex- 
ceed unity.  When  the  numerator  is  equal  to  the  denominator 
the  value  of  the  fraction  is  1  ;  in  every  other  case  the  value 
of  an  improper  fraction  is  greater  than  1 . 

The  following  are  improper  fractions  : 

3        A        6        8.        9         1_2        14        19 

2'      3»      5'       7»     ¥»        6   »        7   »        7   • 

A  Simple  Fraction  is  a  single  expression.     A  simple 
fraction  may  be  either  proper  or  improper. 
The  following  are  simple  fractions : 

13589867 
T»     ^»      6"»      Y»      2»      3^'      ^»     T' 

A  Compound  Fraction  is  a  fraction  of  a  fraction,  or  sev- 
eral fractions  connected  together  with  the  word  of  between 
them. 

Quest. — Write  the  fraction  nineteen-fortieths : — also,  60  fourteeuths — 
18  fiftieths — 16  twentieths — 17  thirtieths — 41  one  thousandths — 85  mil- 
lion ths — 106  fifths.  95.  How  many  kinds  of  vulgar  fractions  are  there? 
What  are  they  ?  What  is  a  proper  fraction  ?  Is  its  value  greater  or  less 
than  1^  What  is  an  improper  fraction?  Why  is  it  called  improper? 
When  is  its  value  equal  to  1  ?  What  is  a  simple  fraction  ?  What  is  a  com- 
pound fraction  ?  Give  an  example  of  a  proper  fraction.  Of  an  improper 
fraction.     Of  a  simple  fraction. 


104  OF    VULGAR    FRACTIONS. 

The  following  are  compound  fractions : 

^ofl    lofiof^,    I  of  3,    iofiof4. 

A  Mixed  Number  is  made  up  of  a  whole  number  and  a 
fraction.    The  whole  numbers  are  sometimes  called  integers, 
'      The  following  are  mixed  numbers  : 

Ql      4.1      a.2      K2_      a5_      o\ 

'^2'»      ^3">      "¥>      *^5»      "¥»      *^T» 

A  Complex  Fraction  is  one  having  a  fraction  or  a  mixed 
number  in  the  numerator  or  denominator,  or  in  both. 
The  following  are  complex  fractions : 

f  2  f        42f 

14'      47i       "f'      87|' 

96.  The  numerator  and  denominator  of  a  fraction,  taken 
together,  are  called  the  terms  of  the  fraction.  Hence,  every 
fraction  has  two  terms. 

97.  A  whole  number  may  be  expressed  fractionally  bv 
writing  1  below  it  for  a  denominator.     Thus, 

3  may  be  written  ^  and  is  read,  3  ones. 

5  "           "        r       "         "  ^  ^^^s. 

6  "            "        f       "         "  ^  o^^s. 
8         "           "        f       "         "  8  ones. 

But  3  ones  are  equal  to  3,  5  ones  to  5,  6  ones  to  6,  and  8 
ones  to  8.  Hence,  the  value  of  a  number  is  not  changed  by 
placing  1  under  it  for  a  denominator. 


Quest. — ^What  is  a  mixed  number?  Give  an  example  of  a  compound 
fraction.  Of  a  mixed  fraction.  Is  four-ninths  a  proper  or  improper  frac- 
tion ?  What  kind  of  a  fraction  is  six-thirds  ?  What  is  its  value  ?  What 
kind  of  a  fraction  is  nine-eighths  ?  What  is  its  value  ?  What  kind-  of  a 
fraction  is  one-half  of  a  third  ?  What  kind  of  a  fraction  is  two  and  one- 
sixth  ?  Four  and  a  seventh  ?  Eight  and  a  tenth  ?  What  is  a  complex 
fraction?  96.  What  are  the  terms  of  a  fraction?  What  are  the  terms  of 
the  fraction  three-fourths?  Of  five -eighths  ?  Of  six-sevenths?  97.  How 
may  a  whole  number  be  expressed  fractionally  ?  Does  this  alter  its  value  1 
Give  an  example. 


OF    VULGAR    FRACTIONS.  105 

98.  If  an  apple  be  divided  into  6  equal  parts, 
^  will  express  one  of  the  parts, 
•|     "         *'        two  of  the  parts, 
■|     "         "         three  of  the  parts, 
&c.,  &c.,  &c., 

and  generally,  the  denominator  shows  into  how  many  equal 
parts  the  unit  is  divided,  and  the  numerator  how  many  of  the 
parts  are  taken. 

Hence,  also,  we  may  conclude  that, 

-i-  X  2  ;  that  is,  ^  taken  2  times  =  |-, 
■1-  X  3  ;  that  is,  ^  taken  3  times  =  |-, 
i  X  4  ;  that  is,  ^  taken  4  times  =  ^, 
&c.,  &c.,  (fee, 

and  consequently  we  have. 

Proposition  I.  If  the  numerator  of  a  fraction  be  multiplied 
by  any  number^  the  denominator  remaining  unchanged^  the  value 
of  the  fraction  will  be  increased  as  many  times  as  there  are  units 
in  the  multiplier.  Hence,  to  multiply  a  fraction  by  a  whole 
number,  we  simply  multiply  the  numerator  by  the  number. 


1.  Multiply  I- by  5. 

2.  Multiply  ^  by  7. 

3.  Multiply  il  by  9. 

4.  Multiply  i|  by  12. 


EXAMPLES. 

5.  Multiply  3-7_  by  11. 

6.  Multiply  ^  by  12. 

7.  Multiply  3-\  by  14. 

8.  Multiply  if  by  15. 


99.  If  three  apples  be  each  divided  into  6  equal  parts, 
there  will  be  18  parts  in  all,  and  these  parts  will  be  expressed 
by  the  fraction  ^s.  if  it  were  required  to  express  but  one- 
third  of  the  parts,  we  should  take,  in  the  numerator,  but  one- 

QuEST. — 98.  If  an  apple  be  divided  into  six  equal  psirts,  how  do  you  ex- 
press one  of  those  parts  ?  Two  of  them  ?  Three  of  them  ?  Four  of  them? 
Five  of  them?  Repeat  the  proposition.  How  do  you  multiply  a  fraction 
by  a  whole  number  ?  99.  If  3  apples  be  each  divided  into  6  equal  parts, 
how  many  parts  in  all  ?  If  4  apples  be  so  divided,  how  many  parts  m  all  ? 
If  5  apples  be  so  divided,  how  many  parts  ?  How  many  parts  in  6  apples  ? 
In  7?  In  8?  In  9?  In  10? 

5* 


106  OF    VULGAR    FRACTIONS. 

third  of  the  eighteen  parts  ;  that  is,  the  fraction  -|  would  ex 
press  one-third  of  ^^^.  If  it  were  required  to  express  one- 
sixth  of  the  18  parts,  we  should  take  one-sixth  of  18,  and  | 
would  be  the  required  fraction. 

In  each  case  the  fraction  Ls  has  been  diminished  as  many 
imes  as  there  were  units  in  the  divisor.     Hence, 

Proposition  II.  If  the  numerator  of  a  fraction  he  divided 
hy  any  number^  the  denominator  remaining  unchanged,  the  value 
of  the  fraction  will  he  diminished  as  many  times  as  there  are 
units  in  the  divisor.  Hence,  a  fraction  may  he  divided  hy  a 
whole  numher  hy  dividing  its  numerator. 


EXAMPLES. 


1.  Divideff  by  6. 

2.  Divide  i||  by  8. 

3.  Divide  ff^  by  12. 

4.  Divide  ifj  by  7. 


5.  Divide  ^^  by  5. 

6.  Divide  f|f  by  12. 

7.  Divide  fff  by  32. 

8.  Divide  ||4  by  36. 


100.  Let  us  again  suppose  the  apple  to  be  divided  into  6 
equal  parts.  If,  now,  each  part  be  divided  into  2  equal  parts, 
there  will  be  12  parts  of  the  apple,  and  consequently  each 
part  will  be  but  half  as  large  as  before. 

Three  parts  in  the  first  case  will  be  expressed  by  |^,  and 
in  the  second  by  ^.     But  since  the  parts  in  the  second  are 
only  half  the  parts  in  the  first  fraction,  it  follows  that, 
^  —  one  half  of  |^. 

If  we  suppose  the  apple  to  be  divided  into  18  equal  parts, 

Quest. — What  expresses  all  the  parts  of  the  three  apples?  What  ex- 
presses one-half  of  them  ?  One-third  of  them  ?  One-sixth  of  them  ?  One- 
ninth  of  them?  One-eighteenth  of  them?  What  expresses  all  the  parts 
of  four  apples  ?  One-half  of  them  ?  One-third  of  them  ?  One-fourth  of 
them  ?  One-sixth  of  them  ?  One-eighth  of  them  ?  One-twelfth  of  them  1 
One-twenty-fourth  of  them?  Put  similar  questions  for  5  apples,  6  apples, 
&c.  Repeat  the  proposition.  How  may  a  fraction  be  divided?  100.  If  a 
unit  be  divided  into  6  equal  parts  and  then  into  12  equal  parts,  how  does  one 
of  the  last  parts  compare  with  one  of  the  first  ?  If  the  second  division  be 
into  18  parts,  how  do  they  compare?    If  into  24? 


OF    VULGAR    FRACTIONS.  107 

three  of  the  parts  will  be  expressed  by  ^,  and  since  the 
parts  are  but  one-third  as  large  as  in  the  first  case,  we  have 

^  =z  one  third  oi  | : 
and  since  the  same  may  be  said  of  all  fractions,  we  have 

Proposition  III.  If  the  denominator  of  a  fraction  he  mul- 
tiplied hy  any  number,  the  numerator  remaining  unchanged,  the 
value  of  the  fraction  will  be  diminished  as  many  times  as  there 
are  units  in  the  multiplier.  Hence,  a  fraction  may  be  divided 
hy  any  number,  by  multiplying  the  denominator  by  that  number. 


EXAMPLES. 

1.  Divide  ||  by  6.  5.  Divide  |f|-  by  14. 

2.  Divide  |f  by  9.  6.  Divide  i^\9  by  15. 

3.  Divide  ^  by  12.  7.  Divide  |||  by  5. 
4  Divide  Hi  by  11.  8.  Divide  At  by  8. 


101.  If  we  suppose  the  apple  to  be  divided  into  3  parts 
instead  of  6,  each  part  will  be  twice  as  large  as  before,  and 
three  of  the  parts  will  be  expressed  by  J  instead  of  |.  But 
this  is  the  same  as  dividing  the  denominator  6  by  2  ;  and 
since  the  same  is  true  of  all  fractions,  we  have 

Proposition  IV.  If  the  denominator  of  a  fraction  he  divi- 
ded by  any  number,  the  numerator  remaining  unchanged,  the 
value  of  the  fraction  will  be  increased  as  many  times  as  there  are 
units  in  the  divisor.  Hence,  a  fraction  may  be  multiplied  by  a 
whole  number,  by  dividing  the  denominator  by  that  number. 


Quest. — What  part  of  24  is  6  ?  If  the  second  division  be  into  80  parts 
how  do  they  compare  ?  If  into  36  parts  ?  Repeat  the  proposition.  How 
may  a  fraction  be  divided  by  a  whole  number  ?  101.  If  we  divide  1  apple 
into  three  equal  parts  and  another  into  6,  how  many  times  greater  will 
the  parts  of  the  first  be  than  those  of  the  second  ?  Are  the  parts  larger 
as  you  decrease  the  denominator  ?  If  you  divide  the  denominator  by  2, 
how  do  you  affect  the  parts  ?  If  you  divide  it  by  3  ?  By  4  ?  By  5  ? 
By  6  ?  By  7  ?  By  8  ?  Repeat  the  proposition.  How  may  a  fraction 
be  multiplied  by  a  whole  number  ? 


108  OF  VULGAR  FRACTIONS. 


EXAMPLES. 


1.  Multiply  I  by  2,  by  4. 

2.  Multiply  16  by  4,  8,  16. 

3.  Multiply  |Aby  4,  6,  12. 

4.  Multiply  m  by  16,  56. 


5.  Multiply  Lsj-  by  7. 

6.  Multiply^ by 5, 10,20. 

7.  Multiply  III  by  8,  by  16. 

8.  Multiply  yf  by  7,  by  21. 


102.  It  appears  from  Prop.  I.  that  if  the  numerator  of  a 
fraction  be  multiplied  by  any  number,  the  value  of  the  frac- 
tion will  be  increased  as  many  times  as  there  are  units  in  the 
multiplier.  It  also  appears  from  Prop.  III.,  that  if  the  de- 
nominator of  a  fraction  be  multiplied  by  any  number,  the  value 
of  the  fraction  will  be  diminished  as  many  times  as  there  are 
units  in  the  multiplier. 

Therefore,  when  the  numerator  and  denominator  of  a  frac- 
tion are  both  multiplied  by  the  same  number,  the  increase 
from  multiplying  the  numerator  will  be  just  equal  to  the  de- 
crease from  multiplying  the  denominator ;  hence  we  have, 

Proposition  V.  If  both  terms  of  a  fraction  be  multiplied 
by  the  same  number,  the  value  of  the  fraction  will  remain  un^ 
changed, 

EXAMPLES. 

1 .  Multiply  the  numerator  and  denominator  of  |-  by  7. 

w    1.  5        5  X  7       35 

We  have,  ■--  = =z  -—. 

7        7  X  7       49 

2.  Multiply  the  numerator  and  denominator  of  ^  by  3,  by 
4,  by  6,  by  7,  by  9,  by  15,  by  17. 

3.  Multiply  both  terms  of  the  fraction  -^^  by  9,  by  12,  by 
16,  by  7,  by  5,  by  11. 

Quest. — 102.  If  the  numerator  of  a  fraction  be  multiplied  by  a  number, 
how  many  times  is  the  fraction  increased  ?  If  the  denominator  bo  multi- 
plied by  the  same  number,  how  many  times  is  the  fraction  diminished  ?  If 
then  the  numerator  and  denominator  be  both  multiplied  at  the  same  time, 
is  the  value  changed  ?     Why  not?     Repeat  the  proposition. 


GREATEST    COMMON    DIVISOR.  109 

103.  It  appears  from  Prop.  II.  that  if  the  numerator  of  a 
fraction  be  divided  by  any  number,  the  value  of  the  fraction 
will  be  diminished  as  many  times  as  there  are  units  in  the  di- 
visor. It  also  appears  from  Prop.  IV.,  that  if  the  denominator 
of  a  fraction  be  divided  by  any  number,  the  value  of  the  frac- 
tion v^^ill  be  increased  as  many  times  as  there  are  units  in  the 
divisor.  Therefore,  when  both  terms  of  a  fraction  are  divided 
by  the  same  number,  the  decrease  from  dividing  the  numerator 
will  be  just  equal  to  the  increase  from  dividing  the  denomina- 
tor :  hence  we  have, 

Proposition  VI.  If  both  terms  of  a  fraction  he  divided 
by  the  same  number,  the  value  of  the  fraction  will  remain  un- 
changed. 

EXAMPLES. 

1.  Divide  both  terms  of  the  fraction  -^-^  by  4:  this  gives 

4)  8     _  2.  An.<i     2. 

4)T16   —4-  ^^^-     4- 

2.  Divide  each  term  by  8  :  this  gives  |j^  =  ■^. 

3.  Divide  each  term  of  the  fraction  -^^  by  2,  by  4,  by  8, 
by  16,  by  32. 

4.  Divide  each  term  of  the  fraction  y^  by  2,  by  3,  by  4, 
by  5,  by  6,  by  10,  by  12,  by  15,  by  20,  by  30,  by  60. 

GREATEST    COMMON    DIVISOR. 

104.  Any  immber  greater  than  unity  that  will  divide  two 
or  more  numbers  without  a  remainder,  is  called  their  com- 
mon divisor :  and  the  greatest  number  that  will  so  divide 
them,  is  called  their  greatest  common  divisor. 

Quest. — 103.  If  the  numerator  of  a  fraction  be  divided  by  a  number, 
how  many  times  will  the  value  of  the  fraction  be  diminished  ?  If  the  de- 
nominator be  divided  by  the  same  number,  how  many  times  will  the  value 
of  the  fraction  be  increased  ?  If  they  are  both  divided  by  the  same  num- 
ber, will  the  value  of  the  fraction  be  changed  ?  Why  not  ?  Repeat  the 
proposition.  104.  What  is  a  common  divisor?  What  is  the  greatest  com- 
mon divisor  of  two  or  more  numbers  ? 


110  GREATEST    COMMON    DIVISOR. 

Before  explaining  the  manner  of  finding  this  divisor,  it  is 
necessary  to  explain  some  principles  on  which  the  method 
depends. 

One  nmnber  is  said  to  be  a  multiple  of  another  when  it 
contains  that  other  an  exact  number  of  times.  Thus,  24  is 
a  multiple  of  6,  because  24  contains  6  an  exact  number  of 
times.  For  a  like  reason  60  is  a  multiple  of  12,  since  it  con 
tains  12  an  exact  number  of  times. 

First  Principle.  Every  number  which  exactly  divides 
another  number  will  also  divide  without  a  remainder  any 
multiple  of  that  number.  For  example,  24  is  divisible  by  8 
giving  a  quotient  3.  Now,  if  24  be  multiplied  by  4,  5,  6,  or 
any  other  number,  the  product  so  arising  will  also  be  divisi- 
ble by  8. 

Second  Principle.  If  a  number  be  separated  into  two 
parts,  any  divisor  which  will  divide  each  of  the  parts  sep- 
arately, without  a  remainder,  will  exactly  divide  the  given 
number.  For,  the  sum  of  the  two  partial  quotients  must  be 
equal  to  the  entire  quotient ;  and  if  they  are  both  whole  num- 
bers, the  entire  quotient  must  be  a  whole  number ;  for  the  sum 
of  two  whole  numbers  cannot  be  equal  to  a  fraction. 

For  example,  if  36  be  separated  into  the  parts  16  and  20, 
the  number  4,  which  will  divide  both  numbers  16  and  20, 
will  also  divide  36  ;  and  the  sum  of  the  quotients  4  and  5  will 
be  equal  to  the  entire  quotient  9. 

Third  Principle.  If  a  number  be  decomposed  into  two 
parts,  then  any  divisor  which  will  divide  the  given  number 
and  one  of  the  parts,  will  also  divide  the  other. 

For,  the  entire  quotient  is  equal  to  the  sum  of  the  two  par- 
tial quotients  ;  and  if  the  entire  quotient  and  one  of  the  partial 
quotients  be  whole  numbers,  the  other  must  also  be  a  whole 
number  ;  for  no  proper  fraction  added  to  a  whole  number  can 
give  a  whole  number. 


Quest. — When  is  one  number  said  to  be  a  multiple  of  another  ?     What 
is  the  first  principle  ?     What  is  the  second  ?     What  is  the  third  ? 


GREATEST    COMMON    DIVISOR.  Ill 

1.  Let  it  be  required  to  find  the       I         operation. 
greatest  common  divisor  of  the  nura-       I  216)408(1 
bers  216  and  408.  j  ^ 

It  is  evident  that  the  greatest  com-       j  192)216(1 

mon  divisor  cannot  be  greater  than      |  __ 

the  least  number  216.    Now,  as  216  24)192(8 

will  divide  itself,  let  us  see  if  it  will      |  ^^"^ 

divide  408 ;  for  if  it  will,  it  is  the  greatest  common  divisor 
sought. 

Making  this  division,  we  find  a  quotient  1  and  a  remainder 
192  ;  hence,  216  is  not  the  greatest  common  divisor.  Now 
we  say,  that  the  greatest  common  divisor  of  the  two  given  num- 
bers is  the  common  divisor  of  the  less  number  216  and  the  re- 
mainder 192  after  the  division.  For,  by  the  second  principle, 
any  number  which  will  exactly  divide  216  and  192,  will  also 
exactly  divide  the  number  408. 

Let  us  now  seek  the  common  divisor  between  216  and 
192.  Dividing  the  greater  by  the  less,  we  have  a  remainder 
of  24  ;  and  from  what  has  been  said  above,  the  greatest  com- 
mon divisor  of  192  and  216  is  the  same  as  the  greatest 
common  divisor  of  192  and  24,  which  we  find  to  be  24. 
Hence,  24  is  the  greatest  common  divisor  of  the  given  num- 
bers 216  and  408  ;  and  to  find  it 

Divide  the  greater  number  by  the  less,  and  then  divide  the 
divisor  by  the  remainder,  and  continue  to  divide  the  last  divisor 
by  the  last  remainder  until  nothing  remains.  The  last  divisor 
will  be  the  greatest  common  divisor  sought. 

EXAMPLES. 

1.  Find  the  greatest  common  divisor  of  408  and  740.  :ar     T 

2.  Find  the  greatest  common  divisor  of  315  and  810.    j»   /f^V^ 

3.  Find  the  greatest  common  divisor  of  4410  and  5670.   c:.  &3  0 

4.  Find  the  greatest  common  divisor  of  3471  and  1869.   s^  triy 

5.  Find  the  greatest  common  divisor  of  1584  and  2772.    ^  j^i  / 

Quest. — Give  the  rule  for  fiiiding  the  ^eatest  common  divisor.  How  do 
you  find  the  greatest  common  divisor  of  more  than  two  numbers  ? 


112  GREATEST    COMMON    DIVISOR. 

Note. — If  it  be  required  to  find  the  greatest  common  divisor  of 
more  than  two  numbers,  find  first  the  greatest  common  divisor  of 
two  of  them,  then  of  that  common  divisor  and  one  of  the  remaining 
numbers,  and  so  on  for  all  the  numbers :  the  last  common  divisor 
will  be  the  greatest  common  divisor  of  all  the  numbers. 

6.  What  is  the  greatest  common  divisor  of  492,  744,  and 
1044?  ^    J^J^-^i  Ans,  • 

7.  What  is  the  greatest  common  divisor  of  944,  1488,  and 

2088  ?       *     7" 

(' 

8.  What  is  the  greatest  common  divisor  of  216,  408,  and 
740?         -    *-| 

9.  What  is  the  greatest  common  divisor  of  945,  1560,  and 
22683  ?      ».  J 

10.  What  is  the  greatest  common  divisor  of  204,  1190, 
1445,  and  2006  ? 


SECOND    METHOD, 

105.  It  has  already  been  remarked  (Art.  90),  that  a  prime 
number  is  one  which  is  only  divisible  by  itself  or  unity,  and 
that  a  composite  number  is  the  product  of  tvs^o  or  more  fac- 
tors (Art.  61).  Now,  every  composite  number  maybe  de- 
composed into  two  or  more  prime  factors.  For  example,  if 
we  have  the  composite  number  36,  we  may  write 

36  =  18x2  =  9x2x2  =  3x3x2x2; 

in  which  we  see  there  are  four  prime  factors,  viz.,  two  3's 
and  two  2's. 

Again,  if  we  have  the  composite  number  150,  we  may 
write 

150  =  15  X  10  =  3x5x  10  =  3x5x5x2; 
in  which  there  are  also  four  prime  factors,  viz.,  one  3,  two 
5's,  and  one  2.     Hence,  to  decompose  a  number  into  its 
prime  factors. 

Quest. — 105.  What  is  a  prime  number?  What  is  a  composite  number? 
Into  what  may  it  b©  decomposed?     What  are  the  prime  factorB  of  36? 


GREATEST    COMMON    DIVISOR. 


1J3 


Divide  it  continually  by  any  prime  number  which  will  divide 
it  without  a  remainder,  and  the  last  quotient,  together  with  the 
several  divisors,  will  be  the  prime  factors  sought. 


EXAMPLES. 

1.  What  are  tlie  prime  factors  of 
180? 

We  first  divide  by  the  prime  num- 
ber 2,  which  gives  90 ;  then  by  3, 
then  by  5,  then  by  3,  and  find  the 
six  prime  factors  2,  3,  5,  3,  and  2. 


2.  What  are  the  prime  factors  of  645  ? 

3.  What  are  the  prime  factors  of  360  ? 


OPERATION. 

2)180 

3)90 

5)30 

3)6 

2 

2x3x5x3x2  =  180 

Ans.  

Ans,  


106.  It  is  plain  that  the  greatest  common  divisor  of  two  or 
more  numbers,  will  always  be  the  greatest  common,  factor, 
and  that  such  factor  must  arise  from  the  product  of  equal 
prime  numbers  in  each.  Hence,  to  find  the  greatest  com- 
mon divisor  of  two  or  more  numbers, 

Decompose  them  into  their  prime  factors,  and  the  product  of 
those  factors  which  are  common  will  be  the  greatest  common 
divisor  sought, 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  1365  and  1995  ? 


3)1365 

5)455 

7)91 

13 

Hence,  3,  5,  7,  and  13  are 

prime  factors. 


3)1995 

5)665 

7)133 

19 

Hence,  3,  5,  7,  and  19  are 

the  factors. 


Hence,   3x5x7  =  1 05  =  the  greatest  common  divisor, 

Quest. — How  do  you  decompose  a  number  into  its  prime  factors?  106. 
What  is  the  greatest  common  divisor  of  two  or  more  numbers  ?  What  does 
such  factor  arise  from  ?   How  then  do  you  find  the  greatest  common  divisor  ? 


114  LEAST    COMMON    MULTIPLE. 

2.  What  is  the  greatest  common  divisor  of  12321  and 
54345  ? 

3.  What  is  the  greatest  common  divisor  of  3775  and  1000  ? 

4.  What  is  the  greatest  common  divisor  of  6327  and 
23997? 

5.  What  is  the  greatest  common  divisor  of  24720  and 
4155? 

LEAST    COMMON    MULTIPLE. 

125.  A  number  is  said  to  be  a  common  multiple  of  two  or 
more  numbers,  when  it  can  be  divided  by  each  of  them,  sepa- 
rately, without  a  remainder. 

The  least  common  multiple  of  two  or  more  numbers,  is  the 
least  number  which  they  will  separately  divide  without  a  re- 
mainder. Thus,  6  is  the  least  common  multiple  of  3  and  2,  it 
being  the  least  number  which  they  will  separately  divide  with 
out  a  remainder. 

A  factor  of  a  number,  is  any  number  greater  than  1  that 
will  di\dde  it  without  a  remainder ;  and  a  prime  factor  is  any 
prime  number  that  will  so  divide  it. 

Now,  it  is  plain,  that  a  dividend  will  contain  its  divisor  an 
exact  number  of  times,  when  it  contains  as  factors,  every  factor 
of  that  divisor :  and  hence,  the  question  of  finding  the  least 
common  multiple  of  several  numbers  is  reduced  to  finding  a 
number  which  shall  contain  all  the  prime  factors  of  each  num- 
ber and  none  others.  If  the  numbers  have  no  common  prime 
factor  their  product  will  be  their  least  common  multiple. 

Example  1. — Let  it  be  required  to  find  the  least  common 
multiple  of  6,  8  and  12. 

125.  When  is  one  number  said  to  be  a  common  multiple  of  two  or 
more  numbers  ?  What  is  the  least  common  multiple  of  two  or  more 
numbers  ?  Of  what  numbers  is  6  the  least  common  multiple  ?  What  is 
the  difference  between  a  common  multiple  and  the  least  common  mul- 
tiple ?  What  is  a  factor  of  a  number  ?  What  is  a  prime  factor  ?  What 
is  a  prime  number  ?  ^Vlien  will  a  dividend  exactly  contain  its  divisor  ? 
To  what  is  the  question  of  finding  the  least  common  multiple  reduced  ? 


LEAST    COMMON    MULTIPLE.  115 


2X3         2X2X2         2X2X8 
6  ....  8    ....  12 


We  see,  from  inspection,  that 
the  prime  factors  of  6,  are  2  and 
3; — of  8;  2,  2  and  2; — and  of 
12;  2,   2  and  3. 

Now,  every  factor,  of  each  number,  must  appear  in  the  least 
common  multiple,  and  none  others :  hence,  we  must  have  all 
the  factors  of  8,  and  such  other  prime  factors  of  6  and  12  as 
are  not  found  among  the  prime  factors  of  8,  that  is,  the  factor  3. 
Hence 

2  X  2  X  2  X  3  =  24,  the  least  common  multiple. 

To  separate  the  prime  factors,  or  to  find  the  least  common 
multiple  of  two  or  more  numbers, 

FIRST   METHOD. 

1.  Place  the  numbers  on  the  same  line,  and  divide  hy  any 
prime  member  that  will  divide  two  or  more  of  them  without 
a  remainder,  and  set  down  in  a  line  below,  the  quotients  and 
the  undivided  numbers, 

n.  Divide  as  before,  until  there  is  no  number  greater  than 
1  that  will  exactly  divide  any  two  of  the  numbers:  then  mul- 
tiply together  the  numbers  of  the  lower  line  and  the  divisors, 
and  the  product  will  be  the  least  common  multiple.  If,  in 
comparing  the  numbers  together,  we  find  no  common  divisor^ 
their  product  is  the  least   common  multiple. 

2.  Find  the  least  common  multiple  of  3,   8,  and  9. 
We  arrange  the  numbers  in  a  line, 

and  see  that  3  will  divide  two  of 
them.  We  then  write  down  the  quo- 
tients 1  and  3,  and  also  the  8  which 
cannot  be  divided.  Then,  as  there 
is  no  common  divisor  between  any  two  of  the  numbers  1,  8? 

Quest. — Give  the  rule  for  finding  the  least  common  multiple.  If  the 
numbers  have  no  common  divisor,  what  is  the  least  common  multiple  ? 


OPERATION. 
3    2X2X2    8X3 
3)3  ..   8  ...  9 

1  ..  8  ...  3 

1X8X3X3  =  72. 


116  LEAST    COMMON    MULTIPLE. 

and  3,  it  follows  that  their  product,  multiplied  by  ihe  divisor  3, 
will  give  the  least  common  multiple  sought. 

3.  Find  the  least  conmion  multiple  of  6,  7,  8,  and  10. 

4.  Find  the  least  common  multiple  of  21  and  49. 

5.  Find  the  least  common  multiple  of  2,  7,  5,  6,  and  8. 

6.  Find  the  least  common  multiple  of  4,  14,  28,  and  98. 

7.  Find  the  least  common  multiple  of  13  and  6. 

8.  Find  the  least  common  multiple  of  12,  4,  and  7. 

9.  Find  the  least  common  multiple  of  6,  9,  4, 14,  and  16. 

10.  Find  the  least  common  multiple  of  13,  12,  and  4. 

11.  What  is  the  least  common  multiple  of  11,  17,  19,  21, 
and  7? 

SECOND    METHOD. 

108.  To  find  the  least  common  multiple  by  this  method. 

Decompose  each  number  into  its  prime  factors ;  after  which, 
select  from  each  number  so  decomposed  the  factors  which  are 
common  to  them  all,  if  there  he  such :  then  select  those  which  are 
common  to  two  or  more  of  the  numbers,  and  so  on  until  all 
the  factors  common  to  any  two  of  them  shall  have  been 
selected.  Then  multiply  these  several  factors  together,  and 
also  the  factors  which  are  not  common,  and  the  product  will 
he  the  least  common  multiple, 

EXAMPLES. 


1.  What  is  the  least  common  multiple  of  99  and  468  ? 

The  prime  factors  of  99 
are  3,  3,  and  11  ;  and  of 
468,  3,  3,  2,  2,  and  13: 
hence,  the  common  factors 
are  3  and  3,  which  are  to  be 
multiplied  by  1 1 , 2, 2,  and  13. 

QuEOT. — 108.  How  do  you  find  the  least  common  multiple  by  the  second 
method? 


OPERATION. 

99=:3x3xll 
468  =  3x3x2x2x13. 

3x3x11x2x2x13=5148. 


REDUCTION  OF  VULGAR  FRACTIONS.        117 


OPERATION. 
12  =:/xXxX 

14=^X7 
36=XxXxXx3. 

2X3x2x3x7  =  252. 


2.  What  is  the  least  common  multiple  of  12,  14,  and  36  ? 
Having     decomposed     the 

numbers  into  their  prime  fac- 
tors, we  see  that  2  is  common 
to  them  all.  We  then  set  it 
aside  as  a  multiplier,  and 
cross  it  in  each  number.  We 
next  set  3  and  2  aside,  and 
cross  them  in  a  contrary  direction.  We  then  have  7  and  3 
remaining,  which  we  use  as  factors.  It  is  plain  that  this 
method  introduces  into  the  common  multiple  every  prime  fac- 
tor of  each  number. 

3.  What  is  the  least  common  multiple  of  4,  9,  10,  15,  18, 
20,  21  ? 

4.  What  is  the  least  common  multiple  of  8,  9,  10,  12,  25, 
32,  75,  80  ? 

5.  What  is  the  least  common  multiple  of  1,2,  3,  4,  5,  6, 
7,8,9? 

6.  What  is  the  least  common  multiple  of  9,  16,  42,  63,  21, 
14,72? 

7.  What  is  the  least  common  multiple  of  7,  15,  21,  28,  35, 
100,125? 

8.  What  is  the  least  common  multiple  of  15,  16,  18,  2Q» 
24,  25,  27,  30  ? 

REDUCTION  OF  VULGAR  FRACTIONS. 

109.  Reduction  of  Vulgar  Fractions  is  the  method  of 
changing  their  forms  without  altering  their  value. 

A  fraction  is  said  to  be  in  its  lowest  terms,  when  there  is 
no  number  greater  than  1  that  will  divide  the  numerator  and 
denominator  without  a  remainder.  The  terms  of  the  fraction 
have  then  no  common  factor. 

Quest. — 109.  What  is  reduction  ?  When  is  a  fraction  said  to  be  in  its 
lowest  terms  ?  Is  one-half  in  its  lowest  terms  ?  Is  the  fraction  two-fourths  ? 
Is  three-fourths? 


118       REDUCTION  OF  VULGAR  FRACTIONS. 


110.  To  reduce  an  improper  fraction  to  its  equivalent 
whole  or  mixed  number. 

Divide  the  numerator  by  the  denominator;  the  quotient  will  be 
the  whole  number ;  and  the  remainder,  if  there  be  one,  placed 
over  the  given  denominator  will  form  the  fractional  part. 

It  was  shown  in  Art.  94,  that  the  value  of  every  fraction 
is  equal  to  the  quotient  arising  from  dividing  the  numerator 
by  the  denominator :  hence  the  value  of  the  fraction  is  not 
changed  by  the  reduction. 

EXAMPLES. 

1.  Reduce  ^  ^^^  V  ^^  *^^^^  equivalent  whole  or  mixed 
numbers. 

OPERATION.  OPERATION. 

4)84  9)67 

Ans,    21  Ans.       7| 

2.  Reduce  ^^  to  a  whole  or  mixed  number.     Ans.  

3.  In  Y  of  yards  of  cloth,  how  many  yards?   Ans.  

4.  In  ^^  of  bushels,  how  many  bushels  ?  Ans.  

5.  If  I  give  \  of  an  apple  to  each  one  of  15  children,  how 
many  apples  do  I  give  ? 

6.  Reduce  fff,  \6_7_2^  V^^  ¥^¥»  ^^  ^^^^^^  whole  or 
mixed  numbers. 

7.  If  I  distribute  878  quarter-apples  among  a  number  of 
boys,  how  many  whole  apples  do  I  use  ? 

8.  Reduce  -^-  to  a  whole  or  mixed  number. 


9.  Reduce  ^-^^  to  a  whole  or  mixed  number. 

10.  Reduce  ^--^■^-  to  a  whole  or  mixed  number. 

11.  Reduce  --|-U--  to  a  whole  or  mixed  number. 


Quest. — 110.  How  do  you  reduce  a  fraction  to  its  equivalent  whole  or 
mixed  number  1  Does  this  reduction  alter  its  value  ?  Why  not  ?  What 
are  four-halves  equal  to  ?  Eight-fourths  ?  Sixteen-eighths  ?  Twenty -fifths  ? 
Thirty -six-sixths  ?  Four-thirds  ?  What  are  nine -fourths  equal  to  ?  Nine- 
fifths  ?     Seventeen -sixths  ?     Eigh teen-sevenths  ? 


REDrCTION  OF  VULGAR  FRACTIONS.        Il9 


111.  To  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

Multiply  the  whole  number  by  the  denominator  of  the  frac' 
tion ;  to  the  product  add  the  numerator ^  and  place  the  sum  over 
the  given  denominator. 

EXAMPLES. 

1 .  Reduce  4^  to  its  equivalent  improper  fraction. 
Here,  4  x  5  =  20  :    then   20  +  4  =  24  ;   hence, 

2-4  is  the  equivalent  fraction. 
This  rule  is  the  reverse  of  Case  I.  In  the  example  4| 
we  have  the  integer  number  4  and  the  fraction  |^.  Now  1 
whole  thing  being  equal  to  5  fifths,  4  whole  things  are  equal 
to  20  fifths ;  to  which,  add  the  4  fifths,  and  we  obtain  the  24 
fifths. 

2.  Reduce  25f  to  its  equivalent  improper  fraction. 

^   _       25x8+3       203     . 
25f  = =—  Ans. 

3.  Reduce  47|-  to  its  equivalent  improper  fraction. 

4.  Reduce  676f^,  8743^3,  690^^,  367^^,  to  their 
equivalent  improper  fractions. 

5.  Reduce  847^?^,  874fl|,  67426|f |,  to  their  equiva- 
lent improper  fractions. 

6.  How  many  200ths  in  675ifJ?  Ans.  

7.  How  many  151ths  in  187^^^^  ^^^-  

8.  Reduce  149|-  to  an  improper  fraction.  Ans,  

9.  Reduce  375f|^  to  an  improper  fraction.  Ans,  

Quest. — 111.  How  do  you  reduce  a  mixed  number  to  its  equivalent  im- 
proper fraction  ?  How  many  fourths  are  there  in  one  ?  In  two  1  In  three  ? 
How  many  sixths  in  four  and  one-sixth  ?  In  eight  and  two-sixths  ?  In 
«even  and  three -sixths  ?  In  nine  and  five -sixths  ?  In  ten  and  five-sixths? 
How  many  eighths  in  two  and  one-eighth  ?  In  three  and  three-eighths  ? 
In  four  and  four-eighths  ?  In  five  and  six  eighths  ?  In  seven  and  seven- 
•ighths?     In  eight  and  seven-eighths? 


120  REDUCTION    OF    VULGAR    FRACTIONS. 

10.  Reduce  17494g|^|^  to  an  improper  fraction. 

11.  Reduce  4834|-|-  to  an  improper  fraction. 

12.  Reduce  1789|-  to  an  improper  fraction. 

13.  Place  4  sevens  in  such  a  manner  that  they  may  be 
equal  to  78. 

CASE    III. 

112.  To  reduce  a  fraction  to  its  lowest  terms. 

I.  Divide  the  numerator  and  denominator  hy  any  number  that 
will  divide  them  both  without  a  remainder,  and  then  divide  the 
quotients  arising  in  the  same  way,  until  there  is  no  number 
greater  than  1  that  will  divide  them  without  a  remainder. 

II.  Or,  find  the  greatest  common  divisor  of  the  numerator  and 
denominator,  and  divide  them  by  it.  The  value  of  the  fraction 
will  not  be  altered  by  the  reduction, 

EXAMPLES. 

1.  Reduce  -^^  to  its  lowest  terms. 

1st  method. 

5)  70       7)14        2        ^.  ^  ^     , 

Jrrzr  =  ry\^  =  "T">  which  are  the  lowest  terms  of  the 

fraction,  since  no  number  greater  than  1  will  divide  the  nu- 
merator and  denominator  without   a  remainder. 

2d  method,  by  the  common  divisor. 

70)175(2 

140  35)    70        2      , 

■ — zzz  — .  Ans. 

Greatest  common  div.     35)70(2  35)  175        5 

70 

2.  Reduce  ^^  to  its  lowest  terms.  Ans,  

3.  Reduce  -g-fff  ^^  i^^  lowest  terms.  Ans,  

Quest. — 112.  When  is  a  fraction  in  its  lowest  terms  ?  (see  Art.  109.)  How 
do  you  reduce  a  fraction  to  its  lowest  terms  by  the  first  method  ?  By  the 
second  ?  What  are  the  lowest  terms  of  two-fourths  ?  Of  six-eighths  ?  Of 
nine-twelfths  ?  Of  sixteen-thirty-sixths  ?  Of  ten-twentieths  ?  Of  fifteen- 
twenty-fourths  ?     Of  sixteen-eighteenths  ?     Of  nine-eighteenths  ? 


REDUCTION  ^F    VULGAR    FRACTIONS  121 

4.  Reduce  I-I-4  to  its  lowest  terms.  Ans.  

4  4  U 

5.  Reduce  -If  J  to  its  lowest  terms.  Ans.  

6.  Reduce  ytis  ^^  ^^^  lowest  terms.  .  •  Ans,  

7.  Reduce  -f^  to  its  lowest  terms  by  the  2d  method. 

8.  Reduce  ^^^  to  its  lowest  terms  by  the  2d  method. 

9.  Reduce  ^^^  to  its  lowest  terms  by  the  2d  method. 

10.  Reduce  ^\  to  its  lowest  terms  by  the  2d  method. 

11.  Reduce  3^%  ^^  ^^^  lowest  terms.  Ans.  

12.  Reduce  |^  to  its  lowest  terms.  Ans,  

13.  Reduce  xyj^-  to  its  lowest  terms.  Ans,  

14.  Reduce  ff^^f  to  its  lowest  terms.  Ans,  

15.  Reduce  y^^  to  its  lowest  terms.  Ans.  • 


113.  To  reduce  a  whole  number  to  an  equivalent  fraction 
having  a  given  denominator. 

Multiply  the  whole  number  hy  the  given  denominator,  and  set 
the  product  over  the  said  denominator. 

EXAMPLES. 

1.  Reduce  6  to  a  fraction  whose  denominator  shall  be  4. 
Here  6  X  4  ==  24 ;  therefore  ^.j*  is  the  required  fraction. 

It  is  plain  that  the  fraction  will  in  all  cases  be  equal  to  the 
whole  number,  since  it  may  be  reduced  to  the  whole  number 
by  Case  I. 

2.  Reduce  15  to  a  fraction  whose  denominator  shall  be  9 

3.  Reduce  139  to  a  fraction  whose  denominator  shall  be 
175. 

4.  Reduce    1837  to   a  fraction  whose  denominator  shall 
be  181. 


Quest. — 113.  HoT^^do  you  reduce  a  whole  number  to  an  equivalent 
fraction  having  a  given  denominator?  How  many  thirds  in  1  ?  In  2  ?  In  3  ? 
In  4  ?  If  the  denominator  be  5,  what  fraction  will  you  form  of  5  ?  Of  4  ? 
Of  9?  Of  7?  Of  8?  With  the  denominator  6,  what  fraction  wUl  you 
form  of  3?    Of  4?    Of  .5  ?    Of  6?    Of  7?    Of  9? 

6 


l^        REDUCTION  OF  VULGAR  FRACTIONS. 

5.  If  the  denominator  be  837,  what  fractions  will  be  formed 
from  327  ?     From  889  ?     From  575  ? 

6.  Reduce- 167  to  a  fraction  whose  denominator  shall 
be  89. 

7.  Reduce  3074  to  a  fraction  whose  denominator  shall 
be  17. 

CASE    V. 

114.  To  reduce  a  compound  fraction  to  its  equivalent  sim- 
ple one. 

I.  Reduce  all  mixed  numbers  to  their  equivalent  improper 
fractions. 

II.  Then  multiply  all  the  numerators  together  for  a  numera^ 
tor  and  all  the  denominators  together  for  a  denominator :  their 
product  will  form  the  fraction  sought, 

EXAMPLES. 

1.  Let  us  take  the  fraction  f  of  f-. 

First,  f  =  3  X  ^ :  hence  the  fractions  may  be  written 
|-of|-  =  3x^off;  that  is,  three  times  one-fourth  of  ^, 
But  -l-  of  I"  =  ^ :  hence  we  have, 

a  result  which  is  obtained  by  multiplying  together  the  nume 
rators  and  denominators  of  the  given  fractions. 

When  the  compound  fraction  consists  of  more  than  two 
simple  ones,  two  of  them  can  be  reduced  to  a  simple  fraction 
as  above,  and  then  this  fraction  may  be  reduced  with  the 
next,  and  so  on.     Hence,  the  reason  of  the  rule  is  manifest. 

2.  Reduce  2}  of.GJ  of  7  to  a  simple  fraction.  Ans,  

Quest. — 114.  What  is  a  compound  fraction  ?  Hot!^  do  you  reduce  a  com- 
pound fraction  to  a  simple  one  ?  Does  this  alter  the  value  of  the  fraction  ? 
What  is  one-half  of  one-half?  One-half  of  one-third  ?  One-third  of  one- 
fourth  ?  One-sixth  of  one-seventh  ?  Three-halves  of  one-eighth  ?  Six-thirds 
of  two-ones? 


REDUCTION  OF  VULGAR  FRACTIONS.        123 

8    Reduce  5  of  J  of  ^  of  6  to  a  simple  fraction. 

METHOD    BY    CANCELLING. 

115.  The  work  may  often  be  abridged  by  striking  out  or 
cancelling  common  factors  in  the  numerator  and  denominator. 

EXAMPLES. 

1.  Reduce  f  of  f  of  |^  to  a  simple  fraction. 

by  cancelling  or  striking  out  the  3's  and  6's  in  the  numerator 
and  denominator. 

By  cancelling  or  striking  out  the  3's  we  only  divide. the 
numerator  and  denominator  of  the  fraction  by  3  ;  and  in  can- 
celling the  6's  we  divide  by  6.  Hence,  the  value  of  the  frac- 
tion is  not  affected  by  striking  out  like  figures j  which  should  al- 
ways be  done  when  they  multiply  the  numerator  and  denominator 

2.  Reduce  |  of  f  of -j^  to  its  simplest  terms. 

2 

Here,  ^x^X^=-^... 

5 
Besides  cancelling  the  like  factors  8  and  8  and  9  and  9,  we 
also  cancel  the  factor  3  common  to  6  and  15,  and  write  the 
quotients  2  and  5  above  and  below  the  numbers. 

3.  Reduce  f  of  |^  of  |  of  ^^j  of  ^  to  its  simplest  terms. 

4.  Reduce  j^  of  ^^  of  i^V  of  f  to  its  simplest  terms. 

5.  Reduce  3|-  of  |  of  ^j  of  49  to  its  simplest  terms. 

CASE    VI. 

116.  To  reduce  complex  fractions  to  simple  ones. 

Reduce  the  numerator  and  denominator,  whe?i  necessary,  to 
simple  fractions :  then  the  numerator  multiplied  by  the  denom- 
inator with  its  terms  inverted,  will  give  the  equivalent  simple 
"^r  action. 

Quest. — 115.  How  may  the  work  often  be  abridged?  116.  What  is  a 
complex  fraction  ?    How  do  you  reduce  a  complex  fraction  to  a  simple  one  1 


18ft  REDUCTION    OF    VULGAR    FRACTIONS, 


EXAMPLES. 
4 

1.  Reduce  the  complex  fraction  ^  to  a  simple  fraction. 

■9 

Now,  if  we  multiply  the. numerator  and  denominator  of  this 
fraction  by  any  number  whatever,  the  value  of  the  fraction 
will  not  be  altered  (Art.  102).  Let  us  then  multiply  them  by 
the  denominator  with  it  terms  inverted.     This  will  give, 


36 

■IJ  —    36 


f  x|-  1  -'"' 

It  is  plain  that  when  the  denominator  is  multiplied  by  the 
fraction  which  arises  from  inverting  its  terms,  the  product 
will  be  equal  to  unity.  Hence,  the  required  simple  fraction 
will  always  be  equal  to  the  numerator  of  the  given  fraction 
multiplied  by  the  denominator  with  its  terms  inverted. 

All  the  cases  in  the  reduction  of  fractions  of  this  class  are 
embraced  in  the  following  eight  forms. 

r,'    .  i         18         8 

Fxrst.  -|.^-x-  =  ^. 

4        ,        8       32       ^. 
Second.        -7-  =  4  X  -  =  —  =  4f . 

Third.  f  =  roX5  =  50- 

4X       4  X  8  +  '7       1       39       13 
Fourth.         -B-^ X5=-=-. 

8f  ~   (72  +  5)   ~  77  ~     /^  77  "~  77  ""11 
~     9  "9 

Sixth.  41  -  I  -  8  ^  9  -  72  -  36' 

5J       ¥       23       9       207       ^,, 
Seventh.       -^  =. -|- :.  _  x  g  ^  —  ^  6^. 

...  ,  ,  H       ¥        88        7        616       308       ^  ,^ 

Eighth.        3!-^ -=yx^=  — =  —  ^2^. 


REDUCTION  OF  VULGAR  FRACTIONS.        125 


2. 

Reduce 

-— ^  to  a  simple  fraction, 
yo 

Ans,  

3 

Reduce 

— ^  to  a  simple  fraction. 

Ans, 

4. 

Reduce 

44 

to  a  simple  fraction. 

147f 

Ans^ 

5. 

Reduce 

247             .      ,    .      .. 
^Y"  to  a  simple  fraction. 

147 

i,"A    to  a  simple  fraction. 

A^^ 

6. 

Reduce 

Ans,  

7. 

Reduce 

8945?7    to  ^  si"^Pl^  fraction. 

CASE    VII. 

Ans.  ^ 

117.  To  reduce  fractions  of  different  denominators  to  equiv- 
alent fractions  having  a  common  denominator. 

I.  Reduce  complex  and  compound  fractions  to  simple  ones, 
and  all  whole  or  mixed  numbers  to  improper  fractions. 

II.  Then  multiply  the  numerator  and  denominator  of  each 
fraction  by  the  product  of  the  denominators  of  all  the  others. 

EXAMPLES. 

1.  Reduce  i,  J,  and  |^  to  a  common  denominator. 
1x3x5  =  15  the  new  numerator  of  the  1st. 
7  X  2  X  5  =  70         "  "  "         2d. 

4  X  3  X  2  =  24         "  "  "         3d. 

and  2x3x5  =  30,  the  common  denominator. 
Therefore,  ^,  |-§,  and  |^  are  the  equivalent  fractions. 
It  is  plain  that  this  reduction  does  not  alter  the  values  ot 
the  several  fractions,  since  the  numerator  and  denominator 
of  each  are  multiplied  by  the  same  number  (see  Prop.  V). 

Quest. — 117.  What  is  the  first  step  in  reducing  fractions  to  a  common 
denominator  ?  What  is  the  second  ?  Does  the  reduction  alter  the  values 
of  the  several  fractions?     Why  not? 


126  REDUCTION    OF    VULGAR    FRACTIONS. 

2.  When  the  numbers  are  small  the  work  may  be  per- 
formed  mentally. 

Thus       i       i        2.  _  2_0        10.        JL6 
XllUb,       21       4>       5    —  40»       4  0»       To- 

Here  we  find  the  first  numerator  by  multiplying  1  by  4  and 
5 ;  the  second,  by  multiplying  1  by  2  and  5  ;  the  third,  by 
multiplying  2  by  4  and  2  ;  and  the  common  denominator  by 
multiplying  2,  4,  and  5  together. 

3.  Reduce  2^  and  ^  of  ^  to  a  common  denominator.  ' 

2^  =  1;    and    lof  1=,^: 
consequently,     ^  and  pj  =  ff  and  ^  are  the  answers. 

4.  Reduce  5^,  |-  of  i,  and  4  to  a  common  denominator. 

5.  Reduce  |-,  i^,  and  37  to  a  common  denominator. 

6.  Reduce  4,  |i,  ^-f  to  a  common  denominator. 

7.  Reduce  7-1-,  ^,  6^  to  a  common  denominator. 

8.  Reduce  4^,  8^,  and  2^  of  5  to  a  common  denomi 
nator. 

9r  Reduce  J,  |-,  f ,  and  ^  to  a  common  denominator. 

10.  Reduce  |^  of  |-  of  |-  and  |-  of  |-  of  f  to  a  common  . 
denominator. 

11.  Reduce  5-|,  3|-,  4i,  and  6|-  to  fractions  having  a  com- 
mon denominator. 

12.  Reduce  f ,  f,  i,  and  J  to  a  common  denominator. 

13.  Reduce  -^,  |-,  f ,  |-,  and  19  to  a  common  denominator. 

14.  Reduce  y^'  g^  24'  sl'  t'  8'  ^^^  ^^  ^^  simple  frac- 
tions having  a  common  denominator. 

118.  It  is  often  convenient  to  reduce  fractions  to  a  common 
denominator  by  multiplying  the  numerator  and  denominator 
in  each  by  such  a  number  as  shall  make  the  denominators 
the  same  in  all. 

Quest. — ^When  the  numbers  are  small,  how  may  the  work  be  perfonned  ? 
118.  By  what  second  method  may  fractions  be  reduced  to  a  common  de- 
nominator? 


REDUCTION  OF  VULGAR  FRACTIONS.        127 
EXAMPLES. 

1.  Let  it  be  required  to  reduce  f  and  -^  to  a  common 
denominator. 

We  see  at  once  that  if  we  multiply  the  numerator  and  de 
nominator  of  the  first  fraction  by  3,  and  the  numerator  and 
denominator  of  the  second  by  2,  they  will  have  a  common 
denominator. 

The  two  fractions  will  be  reduced  to  ^^  and  ^. 

2.  Reduce  ^  and  |-  to  a  common  denominator. 

If  we  multiply  both  terms  of  the  first  fraction  by  3,  and 

both  terms  of  the  second  by  5,  we  have 

±  —  X2.     and    ^—25 
5  —  T5'    ^"^    3  —  T5* 

3.  Reduce  i,  ^,  and  f  to  a  common  denominator. 

4.  Reduce  f ,  ^,  -j^  to  a  common  denominator. 

5.  Reduce  f ,  3|^,  and  |^  to  a  common  denominator. 

6.  Reduce  6^^,  8|^,  and  5^j  to  a  common  denominator. 

7.  Reduce  7^,  |^,  ^,  and  ^  to  a  common  denominator. 

119.  To  reduce  fractions  to  their  least  common  denominator. 

I.  Find  the  least  common  multiple  of  the  denominators  as  in 
Art.  107,  and  it  will  be  the  least  denominator  sought, 

II.  Multiply  the  numerator  of  each  fraction  by  the  quotient 
which  arises  from  dividing  the  common  multiple  by  the  denomi- 
nator, and  the  products  will  be  the  numerators  of  the  required 
fractions ;  under  which  write  the  least  common  multiple, 

EXAMPLES. 

1.  Reduce  f,  f,  and  f  to  their  least  common  denomi- 


and   3  X  4  X  7  X  2  =  168    the  leas 
common  denominator. 


nator. 

OPERATION. 

2)7  .  .  8  .  .  6 

7 

.   .  4  . 

.  3 

Quest. — 119.  How  do  you  reduce  fractions  to  their  least  common  de- 
norainator?     Does  this  reduction  affect  the  values  of  the  fractions? 


128  REDUCTION    OF    DENOMINATE    FRACTIONS. 

168 

-— -  ><:  3  =::  24  X  3  =  72    1st  numerator. 


7 
168 

8 
168 
"6~ 


X  5  =  21  X  5  =  105    2d  numerator. 


X  2  =  28  X  2  =::  56    3d  numerator. 

^^^'  I'-is^m^  and^ 

2.  Reduce  ^,  f,  and  ^3  to  their  least  common  denomi 
nator. 

3.  Reduce  14|-,  6f,  and  5^  to  their  least  common  de 
nominator. 

4.  Reduce  y^^,  2^,  and  |  to  their  least  common  denom- 
inator. 

5.  Reduce  -fi^,  ^,  f  to  their  least  common  denominator. 

6.  Reduce  ^i,  3|^,  41,  and  8  to  a  common  denominator. 

7.  Reduce  3^,  4^^,  8^^,  14^^  to  their  least  common  de- 
nominator. 

8.  Reduce  i,  f,  f,  and  |  to  fractions  having  the  least 
common  denominator. 

9.  Reduce  f ,  -I,  |,  and  -^^  to  fractions  having  the  least 
common  denominator. 

10.  Reduce  ^,  f,  f,  |-,  -^^j  and  ^  to  equivalent  frac- 
tions having  the  least  common  denominator. 

REDUCTION    OF    DENOMINATE    FRACTIONS. 

120.  We  have  seen  (Art.  14),  that  a  denominate  number 
is  one  in  which  the  kind  of  unit  is  denominated  or  expressed. 
For  the  same  reason,  a  denominate  fraction  is  one  which  ex- 
presses the  kind  of  unit  that  has  been  divided.  Such  unit  is 
called  the  unit  of  the  fraction.  Thus,  f  of  a  £  is  a  denomi- 
nate fraction.  It  expresses  that  one  £  is  the  unit  which  has 
been  divided. 

Quest. — 120  What  is  a  denominate  number?  What  is  a  denominate 
fraction  ?  What  is  the  unit  called?  In  two-thirds  of  a  pound,  what  is  the 
unit  of  the  fraction  ? 


REDUCTION    OF    DENOMINATE    FRACTIONS.  129 

The  fraction  f  of  a  shilling  is  also  a  denominate^fraction,  in 
which  the  unit  is  one  shilling.  The  two  fractions,  f  of  a  £ 
and  f  of  a  shilling,  are  of  different  denominations,  the  unit  of 
the  first  being  one  pound,  and  that  of  the  second,  one  shilling. 

Fractions\  therefore,  are  of  the  same  denomination  when  they 
express  parts  of  the  same  unit,  and  of  dijferent  denominations 
when  they  express  parts  of  different  units. 

Redux^tion  of  denominate  fractions  consists  in  changing 
their  denominations  without  altering  their  values. 


121.  To  reduce  a  denominate  fraction  from  a  lower  to  a 
higher  denomination. 

I.  Consider  how  many  units  of  the  given  denomination  make 
one  unit  of  the  next  higher,  and  place  1  over  that  numher  form- 
ing a  second  fraction, 

XL  Then  consider  how  many  units  of  the  second  denomina- 
tion make  one  unit  of  the  denomination  next  higher,  and  place  1 
over  that  number  forming  a  third  fraction,  and  so  on  to  the 
ienomination  to  which  you  would  reduce.  Then  reduce  the  com- 
vound  fraction  to  a  simple  one  (Art.  114). 

EXAMPLES. 

1 .  Reduce  ^  of  a  penny  to  the  fraction  of  a  £, 

OPERATION. 

i  of  3^  of  J^  =  £-r^. 


The  given  fraction  is  ^  of  a 
penny.  But  one  penny  is  equal 
to  Jg-  of  a  shilling :  hence  1  of 
a  penny  is  equal  to  ^  of  ^  of  a  shilling.     But  one  shilling  is 


Quest. — In  three-eightlis  of  a  shilling,  what  is  the  unit?  In  one-half 
of  a  foot,  what  is  the  unit  ?  When  are  fractions  of  the  same  denomination  ? 
When  of  different  denominations  ?  Are  one-third  of  a  £  and  one-fourth 
of  a  jC  of  the  same  or  different  denominations  ?  One-fourth  of  a  .£  and 
one-sixth  of  a  shilling ?  One-fifth  of  a  shilling  and  one-half  of  a  penny? 
What  is  reduction?  How  many  shillings  in  a  jC?  How  many  in  £2} 
In  3  ?  In  4  ?  How  many  pence  in  Is.  ?  In  2  ?  In  3  ?  In  2.^.  8d.  ?  In 
3.9.  Gd.  ?  In  5s.  8d.l  How  many  feet  m  3  yards  2ft.  ?  How  many  inches  ? 
121.  How  do  j'^ou  reduce  a  denominate  fraction  from  a  lower  to  a  higher 
denomination?  What  is  the  first  step?  What  the  second?  What  the  third  I 

6* 


130  REDUCTION    OF    DENOMINATE    FRACTIONS. 

equal  to  J^^  of  a  pound :  hence  ^  of  a  penny  is  equal  to  J  of 
tV  ^^  2V  of  a-  £  =  ^yio"-  '^^^  reason  of  the  rule  is  there- 
fore evident. 

2.  Reduce  f  of  a  barleycorn  to  the  denomination  of  yards. 

OPERATION. 

3  of  1  of  i^^  of  ^  =  ^  yards. 


Since   3   barleycorns 
make  an  inch,  we  first 
place  1  over  3  :  then  as 
12  inches  make  a  foot,  we  place  1  over  12,  and  as  3  feet 
make  a  yard,  we  next  place  1  over  3.      ^ 

3.  Reduce  ^oz.  avoirdupois  to  the  denomination  of  tons. 

4.  Reduce  f  of  a  pint  to  the  fraction  of  a  hogshead. 

5.  Reduce  -j^  of  a  shilling  to  the  fraction  of  a  £. 

6.  Reduce  ^  of  a  farthing  to  the  fraction  of  a  j£. 

7.  Reduce  -|  of  a  gallon  to  the  fraction  of  a  hogshead. 

8.  Reduce  f  of  a  shilling  to  the  fraction  of  a  £. 

9.  Reduce  i|^  of  a  minute  to  the  fraction  of  a  day. 

10.  Reduce  f  of  a  pound  to  the  fraction  of  a  cwU 

11.  Reduce  \  of  an  ounce  to  the  fraction  of  a  ton. 

12.  Reduce  ^-^-^  of  a  farthing  to  the  fraction  of  a  pound. 

13.  Reduce  |-  of  a  penny  to  the  fraction  of  a  pound 

14.  What  part  of  a  Ih.  troy  is  |-  of  a  pwt.  ? 

15.  What  part  of  a  cwt.  is  f  of  a  Z6.  avoirdupois  ? 

16.  What  part  of  a  hhd.  of  wine  is  -^  of  a  gallon  ^ 


122.  To  reduce  a  denominate  fraction  from  a  higher  to  a 
lower  denomination. 

I.  Consider  how  many  units  of  the  next  lower  denomination 
make  1  unit  of  the  given  denomination,  and  place  1  under  that 
number  forming  a  second  fraction. 

II.  Then  consider  how  many  units  of  the  denomination  stiU 
lower  make  one  unit  of  the  second  denomination,  and  place  1 

Quest. — 122.  How  do  you  reduce  a  denominate  fraction  from  a  higher 
to  a  lower  denomination?  What  is  the  first  step?  What  the  second?  What 
the  third  ? 


REDUCTION    OF    DENOMINATE    FRACTIONS.  131 

vador  that  number  forming  a  third  fraction,  and  so  on  to  the 
denomination  to  which  you  would  reduce. 

III.  Connect  all  the  fractions  together,  forming  a  compound 
fraction.  Then  reduce  the  compound  fraction  to  a  simple  one 
(An,.  114.) 

EXAMPLES. 

1 .  Reduce  1  of  a  £  to  the  fraction  of  a  penny. 


OPERATION. 
\   of  2_P  of   I32  ^  iAO^. 


in  this  example  \  of  a  pound 
is  equal  to  l  of  20  shillings.  But 
1  shilling  is  equal  to  12  pence ; 
hence,  i  of  a  £  i=:  i  of  2J>  of  ^^  =  ^^d.    Hence  the  reason 
of  the  rule  is  manifest. 

2.  Reduce  ^cwt,  to  the  fraction  of  a  pound. 

3.  Reduce  -f^  of  a  £  to  the  fraction  of  a  penny. 

4.  Reduce  ^  of  a  day  to  the  fraction  of  a  minute. 

5.  Reduce  f  of  an  acre  to  the  fraction  of  a  pole. 

6.  Reduce  |^  of  a  J£  to  the  fraction  of  a  farthing. 

7.  Reduce  -^-^  of  a  hogshead  to  the  fraction  of  a  gallon. 

8.  Reduce  j^  of  a  bushel  to  the  fraction  of  a  pint. 

9.  Reduce  |^  of  a  day  to  the  fraction  of  a  second. 

10.  Reduce  f  of  a  tun  to  the  fraction  of  a  gill. 

11.  Reduce  f  of  a  pound  to  the  fraction  of  a  farthing. 

12.  Reduce  g|^  of  a  pound  to  the  fraction  of  a  penny. 

13.  Reduce  3^  of  a  lb.  troy  to  the  fraction  of  a  pwt 

14.  Reduce  j^  of  a  cwt.  to  the  fraction  of  a  lb. 

15.  Reduce  -^  of  a  week  to  the  fraction  of  a  second. 

16.  Reduce  f  of  a  ton  to  the  fraction  of  an  ounce. 

17.  Reduce  \^  of  a  yard  to  the  fraction  of  a  nail. 

18.  Reduce  ij  of  a  league  to  the  fraction  of  a  foot. 

19.  Reduce  ^  of  a  lij  to  the  fraction  of  a  scruple. 

Quest. — 123.  How  much  is  one-half  of  a  £1  One-third  of  a  shilling  ? 
One-half  of  a  penny?  How  much  is  one-half  of  a  Ih.  avoirdupois?  One- 
fourth  of  a  ton?  One-fourth  of  a  cwt.1  One-half  of  a  quarter?  One- 
fourth  of  a  quarter?  One-seventh  of  a  quarter?  One-fourteenth  of  a 
quarter?     One -twenty-eighth  of  a  quarter? 


182 


REDUCTION    OF    DENOMINATE    FRACTIONS. 


CASE    III. 

123..  To  find  the  value  of  a  fraction  in  integers  of  a  less 
denomination. 

I.  Reduce  the  numerator  to  the  next  lower  denomination,  and 
then  divide  the  result  hy  the  denominator. 

II.  If  there  be  a  remainder,  reduce  it  to  the  denomination 
still  less,  and  divide  again  by  the  denominator.  Proceed  in  the 
same  way  to  the  lowest  denomination.  The  several  quotients^ 
being  connected  together^  will  form  the  equivalent  denominate 
number. 


EXAMPLES. 


1.  What  is  the  value  of  f  of  a  £  ? 


We  first  bring  the  pounds  to 
shillings.  This  gives  the  frac- 
tion ^^  of  shillings,  which  is  equal 
to  13  shillings  and  1  over.  Redu- 
cing this  to  pence  gives  the  frac- 
tion L2  of  pence,  which  is  equal 
to  4  pence. 


OPERATION. 

2 
20 


3)40 


13^. 


.  .   1 
12 

3)12 


Rem. 


U. 


Ans.  135.  4c?. 


6 
7 
8 
9 
10 
11 


2.  What 

3.  What 

4.  What 

5.  What 
What 
What 
What 
What 
What 
What 


s  the  value  of  ^Ib.  troy  ?  Ans. 

IS  the  value  of  -^-^  of  a  cwt.  ?  Ans. 

Ls  the  value  of  |-  of  an  acre  ?  Ans. 

s  the  value  of  -i-  of  a  £  ?  Ans. 

is  the  value  of  |-  of  a  hogshead  ?  Ans. 
is  the  value  of  i||-  of  a  hogshead  ?  Ans. 
s  the  value  of  |-  of  a  guinea  ?  Ans, 

LS  the  value  of  f  of  a  lb.  troy  ?  Ans. 

s  the  value  of  |-  of  a  tun  of  wine  1  Ans. 
IS  the  value  of  •!•  of  f  of  a  lb.  troy  ?  Ans. 


Quest. — How  do  you  find  the  value  of  a  frajtion  in  terms  of  integers  of 
a  less  denomination  ? 


REDUCTION    OF    DENOMINATE    FRACTIONS.  133 

12.  What  is  the  value  of  f  of  a  league  1  Ans. 

13.  What  is  the  value  of  f  of  |  of  an  acre  ?     Ans.  

14.  What  is  the  value  of  |  of  15  yards  of  cloth  ? 

15.  What  is  the  value  of  f  of  a  tun  of  wine  1 

16.  What  is  the  value  of  j\  of  a  butt  of  beer  ? 

17.  What  is  the  value  of  -^-^  of  a  year  ? 

18.  What  is  the  value  of  f  of  a  chaldron  of  coal? 

19.  What  is  the  value  of  |  of  135.  Ad.  ? 

20.  What  is  the  value  of  f  of  l^cwt.  3qr,  14lb,  ? 

21.  What  is  the  value  of  f  of  a  cubic  yard  ? 

22.  What  quantity  of  ale  is  contained  in  f  of  15228  cubic 
inches,  English  measure  ? 

CASE    IV. 

124.  To  reduce  a  denominate  number  to  a  denominate 
fraction  of  a  given  denomination. 

Reduce  the  number  to  the  lowest  denomination  mentioned  in 
it :  then  if  the  reduction  is  to  be  made  to  a  denomination  still 
less,  reduce  as  in  Case  IL  ;  but  if  to  a  higher  denomination^ 
reduce  as  in  Case  I. 

EXAMPLES. 

1.  Reduce  4^.  Id.  to  the  fraction  of  a  j£. 
We  first  reduce  the 

given    number    to   the 
lowest       denomination 
named  in  it,  viz.,  pence. 
Then,  as  the  reduction  is  to  be  made  to  pounds,  a  higher  de- 
nomination, we  reduce  by  Case  I. 

2.  What  part  of  a  bushel  is  2pk.  3qt.  1 


OPERATION. 

45.  7d.  =  55d. 
Then,  55  of  T^  of  ^  zrz  £^. 


We  first  reduce  to  quarts,  this 
being  the  lowest  denomination. 
We  then  reduce  to  bushels  by 
Case  I. 


OPERATION. 

2pk.  3qt.  =  I9qt, 
19  of!  of  i  =  i|6w 


Quest. — 124.  How  do  you  reduce  a  denominate  number  to  a  fraction 
of  a  given  denomination? 


134  REDUCTION    OF    DENOMINATE    FRACTIONS. 

3    Reduce  2  feet  2  inches  to  the  fraction  of  a  yard. 

Ans, 

4.  Reduce  3  gallons  2  quarts  to  the  fraction  of  a  hogshead. 

Ans, 

5.  Reduce  Iqr.  lib.  to  the  fraction  of  a  hundred. 

Ans.       cwt, 

6.  What  part  of  a  hogshead  is  '^qt.  \pt.  1  Ans.  

7.  What  part  of  a  mile  is  Qft.  7 in.  1  Ans,  

8.  What  part  of  a  mile  is  1  inch  ?  Ans.  

9    What  part  of  a  month  of  30  days,  is  1  hour  1  minute 

1  second?  Ans.  

10.  What  part  of  1  day  is  3hr.  3m,  1  Ans.  

1 1 .  What  part  is  3hr.  3m.  of  2  days  ?    Of  3  ?    Of  4  ?    Of 
10?     Of25? 

12.  Reduce  15^.  lid.  to  the  fraction  of  a  pound. 

13.  Reduce  5^d.  to  the  fraction  of  a  shilling. 

14.  Reduce  Icwt.  2qr.  6lb.  3oz.  S^dr.  to  the  fraction  of  a 
cwt. 

15.  Reduce  boz.  3\gr.  to  the  fraction  of  a  Ih.  troy. 

16.  Reduce  3qr.  3\na.  to  the  fraction  of  an  English  ell. 

17.  Reduce  l^lda.  Ibhr.  to  the  fraction  of  a  year. 

18.  What  part  of  a  pound  is  15^.  9^d.  1 

19.  What  part  of  a  groat  is  ^  of  three  halfpence  ? 

20.  Reduce  4:bu.  2^pk.  of  corn  to  the  fraction  of  a  quarter. 

21.  Reduce  Iqr.  3na.  to  the  fraction  of  a  yard. 

22.  Reduce  2R.  15P.  to  the  fraction  of  an  acre. 

23.  Reduce  23  ll^r.  to  the  fraction  of  a  Rj. 

24.  Reduce  3^'^.  Ipt,  2gi,  to  the  fraction  of  a  hogshead. 

25.  Reduce  184  cubic  inches  to  the  fraction  of  a  cubic 
yard. 

26.  Reduce  I7bu,  3pk,  to  the  fraction  of  a  London  chal- 
dron. 

27.  Reduce  24^  33^^  to  the  fraction  of  a  degree. 

28.  Reduce  27gal.  3qt.  Ipt.  to  the  fraction  of  a  hogshead, 
beer  measure 


ADDITION    OF    VULGAR    FRACTIONS.  135 

ADDITION  OF  VULGAR  FRACTIONS. 

Addition  of  integer  numbers  is  the  process  of  finding  a  sin- 
gle number  which  shall  express  all  the  units  of  the  numbers 
added  {Art.  48.) 

Addition  of  fractions  is  the  process  of  finding  a  single  frac- 
tion which  shall  express  the  value  of  all  the  fractions  added. 

It  is  plain  that  we  cannot  add  fractions  so  long  as  they 
have  different  units  :  for,  i  of  a  £  and  i  of  a  shilling  make 
neither  £1  nor  1  shilling. 

Neither  can  we  add  parts  of  the  same  unit  unless  they  are 
like  parts ;  for  i  of  a  £  and  i  of  a  £  make  neither  f  of  a  £ 
nor  |-  of  a  £.  But  i  of  a  £  and  ^  of  a  £  may  be  added  : 
they  make  f  of  a  £.  So,  J  of  a  £  and  f  of  a  £  make  | 
of  a  £. 

Hence,  before  fractions  can  be  added,  two  things  are 
necessary. 

1st.  That  the  fractions  be  reduced  to  the  same  denomi" 
nation;  that  is,   to  the  same  unit: 

2d.  That  they  he  reduced  to  a  common  denominator; 
that  is,   to  the  same  fractional  unit  (Art.  94). 


126.  When  the  fractions  to  be  added  are  of  the  same  de- 
nomination and  have  a  common  denominator. 

Add  the  numerators  together,  and  place  their  sum  over  the 
common  denominator :  then  -  reduce  the  fraction  to  its  lowest 
terms,  or  to  its  equivalent  mixed  number. 

Quest. — 125.  What  is  addition  of  integer  numbers  ?  What  is  addi- 
tion of  fractions  ?  What  two  things  are  necessaiy  before  fractions  can 
be  added  ?  Can  one-half  of  a  £  be  added  to  one-half  ot  a  shilling 
without  reduction  ?  Can  one-half  be  added  to  one-fourth  without  reduc- 
tion? 126.  When  the  fractions  are  of  the  same  denomination  and 
have  a  common  denominator,  how  do  you  find  their  sum  ?  What  is  the 
sum  of  one  third  and  two-thirds  ?  Of  three-fourths,  one-fourth,  and  four- 
fourths?  Of  three -fifths,  six-fifths,  end  two-fifths?  Of  three-sixths,  seveu- 
Bixtlis,  and  nine-sixths?     Of  oive-eighth,  three-eighths,  and  four-eighths? 


136 


ADDITION    OF    VULGAR    FRACTIONS. 


OPERATION. 

1  +  3  +  6+.3  =  13; 
hence,  ^^  =  sum. 


EXAMPLES. 

1.  Add  i,  |-,  "I,  and  |  together. 
It  is  evident,  since  all  the  parts 

are  halves,  that  the  true  sum  will 
be  expressed  by  the  number  of 
halves  ;  that  is,  by  thirteen  halves. 

2.  Add  i  of  a  £,  |-  of  a  £,  and  |-  of  a  £  together. 

3.  What  is  the  sum  of  f  +  |-  +  f  +  ^^  +  L6  ? 

4.  What  is  the  sum  of  ^^  +  _8_  +  _9_  4.  ^  +  _^? 

5.  What  is  the  sum  of  f  +  f  +  y  +  V  +  V  ? 

6.  What  is  the  sum  of  j-\  +  -^  +  ^Q-  -f-  1 0  _{-  1 3  4.  17  ? 


127.  When  the  fractions  are  of  the  same  denomination  but 
have  different  denominators. 

Reduce  complex  and  compound  fractions  to  simple  ones,  mixea 
numbers  to  improper  fractions,  and  all  the  fractions  to  a  com- 
mon denominator.     Then  add  them  as  in  Case  I, 


EXAMPLES. 


1.  Add  I",  ^,  and  f  together. 

By  reducing  to  a  com- 
mon denominator,  the  new- 
fractions  are 


9.0.     I     AO.  4-  I_2  142 

30'30'^30   —     30» 

which,   by  reducing  to   the 
lowest  terms  becomes  4^^. 


OPERATION. 
6  X  3  X  5  —  90   1st  numerator 
4  X  2  X  5  —  40  2d  numerator. 
2X3X2  =  12  3d  numerator. 
2x3x5z=r30  the  denominator 


2.  Add  i  of  a  jC,  f  of  a  £,  and  |-  of  a  £  together. 

3.  Add  together  \,  ^,  4i,  and  61-.  Ans. 


T»    ¥»    ^3"'    """^    ^5* 

4.  Find  the  least  common  denominator  (Art.  119)  and  add 
the  fractions  y^g-,  ^,  |-,  and  |-.  Ans.  


Quest. — 127.  How  do  you  add  fractions  which  have  different  denomi- 
nators ?  How  do  you  reduce  fractions  of  different  denominators  to  equiva- 
lent fractions  having  a  common  denominator  ? 


ADDITION    OF    VULGAR    FRACTIONS.  137 

5.  Find  the  least  common  denominator  and  add  j^,  |,  -|, 


and  ^.  Ans.  

6.  Find  the  least  common  denominator  and  add  §  of  f, 
I  of  19,  and  f  of  12  together.  Ans,  

7.  Add  f,  ^  ofj\  off,  and  f  of  f  of  11  together. 

128.  When  there  are  mixed  numbers,  instead  of  reducing 
them  to  improper  fractions  we  may  add  the  whole  numbers 
and  the  fractional  parts  separately  and  then  add  their  sums. 

EXAMPLES. 

1.  Add  19-1-,  6f,  and  4|-  together. 

OPERATION. 

Whole  numbers.  Fractional  parts. 

19  +  6  +  4  :.=  29.  i  +  2   +   4  ^  16|  ^  1^^. 

Hence,         29  +  Ij^^  =  ^^to%^   the  sum. 

2.  Add  together  31    6f ,  S-^^,  and  65|.  Aiis.  

3.  Add  together  |^,  |^,  13,  and  ISy^^^,  Ans,  

4.  Add  together  -^q,  \p,  1,  and   L6.  ^.;^^.  

5.  Add  together  38|^,  13^,  and  9f.  Ans.  

6.  Add  together  Gf,  13f-,  18^^,  and  132|.     Ans.  

7.  Add  3|-,  4|-,  and  ^  together.  Ans.  — — 

8.  Add  f  and  -^q  of  j^j  of  15|  together.  Ans.  

45  47— 

9.  Add  i,  7|,  ——-,  and  — -^  together.        Ans.  

CASE    III. 

129.  When  the  fractions  are  of  different  denominations. 

Reduce  the  fractions  to  the  same  denomination.  Then  re- 
dude  all  the  fractions  to  a  common  denominator ,  and  then  add 
them  as  in  Case  I. 

Quest. — 128.  How  may  you  proceed  when  there  are  mixed  numberp? 
129.  When  the  fractions  are  of  different  denominations,  how  are  they 
added  ?     What  is  the  second  method '? 


138  ADDITION    OF    VULGAR    FRACTIONS. 

EXAMPLES. 

1 .  Add  f  of  a  £  to  -I  of  a  shilling 

f  of  a  £  HIT  I  of  2^0  =  ^/  of  a  shilling. 
Then,    4_o  ^.  s  ^  2_^o  _^  is  ^  2_^5^^  _  8_5^^  ^  14^^  ^d. 

Or,  the  1^  of  a  shilling  might  have  been  reduced  to  the  frac- 
tion of  a  £  thus, 

f  of2V  =  Tfo.ofa£=Jjofa£. 

Then,  f  +  2V  =  ff  +  "^=yi®^^  ^-  which  being  re- 
duced gives  14^.  2d. 

2.  Add  f  of  a  yard  to  f  of  an  inch. 

3.  Add  i  of  a  v^reek,  ^  of  a  day,  and  i  of  an  hour  together. 

4.  Add  -|-  of  a  cwt.,  8|^ZZ>.,  and  ^yqO^-  together. 

5.  Add  1^  miles,  ^^  furlongs,  and  30  rods  together. 

Note.— The  value  of  each  of  the  fractions  may  be  found  sepa- 
rately, and  their  several  values  then  added. 

6.  Add  f  of  a  year,  i  of  a  week,  and  I-  of  a  day  together. 
■|  of  a  year  =z  |-  of  ^^  days  r=  219  days. 

J  of  a  week  =  i  of  7  days      z=z      2  days  8  hours. 
I"  of  a  day     =  .         .         .         -         3  hours. 

Ans.  221  da.     llhr. 


7.  Add  f  of  a  yard,  |  of  a  foot,  and  |-  of  a  mile  together. 

8.  Add  f  of  a  cwt.,  ^  of  a  lb.  ISoz.,  and  1  of  a  cwt.  6lb. 
together. 

9.  Add  f  of  a  pound,  f  of  a  shilling,  and  I-  of  a  penny 
together. 

10.  What  is  the  sum  of  f  of  £1  10^.,  \  of  £1  10^.,  and 
^  of  a  hundred  guineas  ? 

11.  Add  3"  of  a  lb.  troy  to  i  of  an  ounce.  Ans.  

12.  Add  1^  of  a  ton  to  ^^  of  a  cwt.  Ans.  — — 


13.  Add  f  of  3  ells  English  to  ^^  of  a  yard.     Ans.  

14.  Add  I"  of  a  yard,  |-  of  a  foot,  and  ^j  of  a  mile  together. 

15.  Add  f  of  an  acre,  f  of  19  square  feet,  and  |-  of  a  squaro 
inch  together. 


SUBTRACTION    OF    VULGAR    FRACTIONS.  139 

16.  What  is  the  sum  of  f  of  a  tun  of  wine  and  f  of  a  hhd,  ? 

17.  Add  I  of  a  chaldron  to  |-  of  a  bushel. 

18  Add  \  of  a  week,  J  of  a  day,  and  \  of  an  hour  to- 
gether. 

19.  Add  ^  of  I  of  a  year,  |  of  ^  of  a  day,  and  §  of  |  of 
19^  hours  together. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

130.  It  .has  been  shown  (Art.  125),  that  before  fractions 
can  be  added  together  they  must  be  reduced  to  the  same  unit 
and  to  a  common  denominator.  The  same  reductions  must 
be  made  before  subtraction. 

Subtraction  of  Vulgar  Fractions  is  the  process  of  finding 
the  difference   between  two  fractional  numbers, 

CASE    I. 

131.  When  the  fractions  are  of  the  same  denomination 
and  have  a  common  denominator. 

Subtract  the  less  numerator  from  the  greater,  and  place  the 
difference  over  the  common  denomisiator, 

EXAMPLES. 

1 .  What  is  the  difference  between  |-  and  f  ? 

Here  we  have  5  —  3  =  2:  hence,  f  =  the  difference. 

2.  From  f|f  take  l|f.  Ans,  

3.  From  ||H  take  ^^,  Ans,  

4.  From  i||f^  take  |^f .  Ans,  

CASE    II. 

132.  When  the  fractions  are  of  the  same  denomination, 
but  have  different  denominators. 

Quest. — 130.  Can  one-third  of  a  shilling  be  subtracted  from  one-third 
of  a  £  without  reduction  ?  Can  one-fourth  of  a  shilling  be  subtracted  from 
one-fifth  of  a  shilling?  What  reductions  are  necessary  h|ibre  subtraction? 
W^hat  is  subtraction?  131.  How  do  you  subtract  fractions  of  the  same 
denomination  and  denominator  ? 


140       SUBTRACTION  OF  VULGAR  FRACTIONS. 

Reduce  mixed  numbers  to  improper  fractions,  compound  and 
complex  fractions  to  simple  ones,  and  all  the  fractions  to  a  com' 
man  denominator :  then  subtract  them  as  in  Case  I, 

EXAMPLES. 

1.  What  is  the  diiFerence  between  |-  and  i? 
Here,      '  ^  ^^  =  ^-^=^  =  ^  answer. 

2.  What  is  the  difference  between  12^  of  i  and  2  ? 

3.  What  is  the  difference  between  2J  of  a  £,  and  ^  of 
a£?  • 

4.  From  i  of  6,  take  ff  of  i.  Ans.  

5.  From  i  of  f  of  7,  take  |  of  f .  Ans.  

6.  From  37ii,  take  3f  of  i.  ^n^.  


7.  What  is  the  difference  between  f  and  j^g-?  Ans. 

8.  What  is  the  difference  between  3|-  and  f  of  f  ? 

495.  343. 

■  9.  What  is  the  difference  between  -— f  and  — -~  ? 

97  146y^y 

10.  From  115f  take  39-|.  Ans,  


1 1 .  Subtract  -^^^  from  a  unit.  il^z^.  

12.  Subtract  ii  from  365.  Ans.  

13.  What  is  the  difference  between  |-  of  15  and  ^  of  72  ? 

14.  To  what  fraction  must  I  add  f  that  the  sum  may  be  |  ? 

15.  What  number  is  that  to  which  if  7f  be  added,  the  sum 
will  be  17f  ? 

16.  What  number  is  that  from  which  if  you  subtract  ^  of 
I  of  a  unit,  and  to  the  remainder  add  f  of  -I  of  a  unit,  the  sura 
will  be  9  ? 

CASE    III. 

133.  When  the  fractions  are  of  different  denominations. 

Reduce  the  fractions  to  the  same  denomination.  Then  re 
duce  them  to  a  common  denominator ;  after  which  subtract  as 
in  Case  I. 

Quest. — 132.  How  do  you  subtract  fractions  of  different  denominators  1 
What  is  the  difference  between  one-half  and  one-third?  133.  How  do  you 
subtract  fractions  which  are  of  different  denominations  ? 


MULTIPLICATION  OF  VULGAR  FRACTIONS.     141 

EXAMPLES. 

1.  What  is  the  difference  between  i  of  a  £  and  J  of  a 
shilling  ? 

i  of  a  shilling  =  1  of  ^  =  Jq  of  a  £>. 

2.  What  is  the  difference  between  -g-  of  a  day  and  f  of  a 
second  ? 

3.  From  1|  of  a  Ih.^  troy  weight,  take  \  of  an  ounce. 

4.  What  is  the  difference  between  -j^  of  a  hogshead  and 
^  of  a  quart  ? 

5.  From  -i^  of  a  £  take  f  of  a  shilling.  Ans,  

6.  From  ^oz.  take  \pwt.  Ans.  — — 

7.  From  A^cwt.  take  ^Yo^h,  Ans.  

8.  What  is  the  difference  between  f  of  a  pound  and  |-  of  a 
shilling  ? 

9.  From  f  of  a  Ih.  troy  take  f  of  an  ounce.    Ans.  —^ — 

10.  From  f  of  a  ton  take  f  of  |  of  a  Ih.  Ans,  

1 1 .  From  |-  of  f  of  a  hhd.  of  wine  take  |-  of  -J-  of  a  pint, 

12.  From  f  of  a  league  take  f  of  a  mile.  Ans.  

13.  From  f  of  365^  days  take  f  of  ^  of  an  hour. 

14.  A  pound  avoirdupois  is  equal  to  \Aoz.  Wpwt.  \Qgr, 
Iroy;  what  is  the  difference,  in  troy  weight,  between  the 
ounce  avoirdupois  and  the  ounce  troy  I 

MULTIPLICATION  OF  VULGAR  FRACTIONS. 

134.  Multiplication  is  a  short  method  of  taking  one  num- 
ber, called  the  multiplicand,  as  many  times  as  there  are  units 
in  another  number,  called  the  multiplier. 

Hence,  when  the  multiplier  is  less  than  1  we  do  not  take  the 
whole  of  the  multiplicand,  but  only  such  a  part  of  it  as  the  mul- 


Quest. — 134.  What  is  multiplication  ?  What  is  required  when  the  mul- 
tiplier is  less  than  1?  Does  multiplication  then  imply  increase?  What 
is  the  product  of  8  multiplied  by  one-half?  By  one-fourth?  By  one- 
eighth?  By  three-halves?  By  six-halves?  What  is  the  product  of  9 
multiplied  by  one-half?     By  one-third?     By  one-sixth?     By  one-ninth? 


142  MULTIPLICATION    OF    VULGAR    FRACTIONS. 

tiplier  is  of  unity.  For  example,  if  the  multiplier  be  one-half 
of  unity,  the  product  will  be  half  the  multiplicand ;  as,  for  ex- 
ample, the  product  of  8  multiplied  by  -J  is  4.  If  the  multi- 
plier be  i  of  unity,  the  product  will  be  one-third  of  the  mul 
tiplicand.  Hence,  to  multiply  by  a  proper  fraction  does  not 
imply  increase  J  as  in  the  multiplication  of  whole  numbers. 

CASE    I, 

135.  To  multiply  a  fraction  by  a  whole  number. 

Multiply  the  numerator,  or  divide   the  denominator  by  the 
whole  number, 

EXAMPLES 


OPERATION. 

or  by  dividing  the  denom- 
inator by  4,  we  have 

5    \^   A    —       5   —   5    oj 


1.  Multiply  the  fraction  |-  by  4. 

When  it  is  required  to  mul- 
tiply a  fraction  by  a  whole 
number,  it  is  required  to  in- 
crease the  fraction  as  many 
times  as  there  are  units  in  the 
multiplier,  which  may  be  done 
by  multiplying  the  numerator  (Art.  98),  or  by  dividing  the 
denominator  (Art.  101). 

2.  Multiply  -^  by  12.  Ans.  

3.  Multiply  11-  by  7.  Ans,  

4.  Multiply  Yt  by  9.  Ans.  

5.  Multiply  ij-2_7  by  5.  j^j^g    

6.  Multiply  fff  by  49.  Ans.  

7.  Multiply  f|i  by  357.  Ans,  

8.  Multiply  iffl  by  198.  Ans,  

9.  Multiply  gyWe  ^y  2433.  Ans.  

Quest. — When  the  multiplier  is  less  than  1,  how  much  of  the  multipli- 
cand is  taken  ?  Does  the  multiplication  by  a  proper  fraction  imply  increase  ? 
135.  How  do  you  multiply  a  fraction  by  a  whole  number?  136.  What  is 
the  product  of  one-sixth  by  one-seventh?  Of  three-fourths  by  one-half? 
Of  six-ninths  by  three-fifths  ?  Give  the  general  rule  for  the  multiplication 
of  fractions. 


MULTIPLICATION    OF    VULGAR    FRACTIONS.  143 

CASE    II. 

136.  To  multiply  one  fraction  by  another. 

Reduce  all  the  mixed  numbers  to  improper  fractions,  and  all 
compound  and  complex  fractions  to  simple  ones :  then  multiply 
the  numerators  together  for  a  numerator,  and  the  denominators 
together  for  a  denominator. 


OPERATION. 

fxf=fx5xi=M 


EXAMPLES. 

1.  Multiply  |by  f. 
In  this  example  \  is  to  be 

taken  |-  times ;  that  is,  |-  is  first 

to  be  multiplied  by  5  and  the 

product  divided  by  7,  a  result  which  is  obtained  by  multiply 

ing  the  numerators  and  denominators  together. 

2.  Multiply  i  of  f  by  8^. 

We  f\rst  reduce  the  com- 
pound fraction  to  the  simple 
one  ^,  and  the  mixed  num- 


OPERATION. 
1  of  3   —     3 

fti  —  25 


Hence^^O<^  =  T^=||. 


ber  to  the  equivalent  fraction 
^  ;  after  which,  we  multi- 
ply the  numerators  and  denominators  together. 

3.  Multiply  51  by  i.  Ans.  

4.  Multiply  12  by  I  of  9.  Ans.  

5.  Multiply  i  of  3  of  -i  by  \^.  Ans.  

6.  Multiply  f  by  f  of  f  Ans,  

7.  Required  the  product  of  6  by  ^  of  5.  Ans.  

8.  Required  the  product  of  |-  of  f  by  f  of  3|-. 

9.  Required  the  product  of  3|-  by  4^^ .  Ans.  

10.  Required  the  product  of  5,  f,  ^  off,  and  4i. 

11.  Required  the  product  of  4^,  |-  of  i,  and  18|. 

12.  Required  the  product  of  14,  |-,  f  of  9,  and  6-|. 

13.  What  is  the  product  of  16f ,  ^,   1    X,  and  -^  \ 

3  2  1 6 


144 


MULTIPLICATION    OF    VULGAR    FRACTIONS. 


137.  In  multiplying  by  a  mixed  number,  we  may  first  mul- 
tiply by  the  integer,  then  multiply  by  the  fraction,  and  then 
add  the  two  products  together.  This  is  the  best  method 
when  the  numerator  of  the  fraction  is  1. 


EXAMPLES. 


1.  Multiply  26  by  3i. 


We  first  multiply  26  by  3:  the 
product  is  78.  Afterwards  we  mul- 
tiply 26  by  i:  the  product  is  13: 
hence  the  entire  product  is  91. 

2.  Multiply  48  by  81. 

We  first  multiply  by  8,  and  then 
add  a  sixth. 

3.  Multiply  67  by  9^. 

4.  Multiply  842  by  7J. 

5.  Multiply  3756 -by  3J. 

6.  Multiply  2056  by  5^. 


OPERATION. 
26 

3 

78 


26  xi 

=  13 

91 

OPERATION. 

48 

X  8  = 

384 

48 

xi- 

8 

392. 

Arts, 
Arts. 
Ans. 
Arts, 


GENERAL    EXAMPLES. 


1.  What  is  the  product  of  |  of  f ,  f  of  15^,  and  /_  of  2  ? 


2.  What  is  the  continued  product  of  14f , 

6 

4      71      f 
51.'     15'      f 

and   ^? 

Ans.  • 

3.  What  is  the   product  of   -|-,    ^,    -/, 

4f       ,      , 

and  20  ? 

Ans.  

4.  What  is  the  product  of  f  of  ^7_  of  15,  and  i|  of  llf  ? 

Ans.  

5.  What  will  7  yards  of  cloth  cost,  at  $-|  per  yard  ? 

Quest. — 137.  How  may  you  multiply  by  a  mixed  number  ?    When  \% 
this  the  best  method  ? 


DIVISION    OF    VULGAR    FRACTIONS.  145 

6    What  will  32  gallons  of  brandy  cost,  at  $1|-  per  gallon  '^ 

7.  If  lib,  of  tea  cost  $1^,  what  will  61/6.  cost? 

8.  What  will  be  the  cost  of  17^  yards  of  cambric,  at  2^ 
shillings  per  yard  1 

9.  What  will  15j3g.  barrels  of  cider  come  to,  at  $3  per 
barrel  1 

10.  What  will  3f  boxes  of  raisins  cost,  at  $21  per  box  1 

11.  What  will  15i  barrels  of  sugar  cost,  at  171  dollars  per 
barrel? 

DIVISION  OF  VULGAR  FRACTIONS. 

138.  We  have  seen  that  division  of  entire  numbers  ex- 
plains the  manner  of  finding  how  many  tidies  a  less  number 
is  contained  in  a  greater. 

In  division  of  fractions  the  divisor  may  be  larger  than  the 
dividend,  in  which  case  the  quotient  will  be  less  than  1. 

For  example,  divide  1  apple  into  4  equal  parts. 

Here  it  is  plain  that  each  part  will  be  -J- ;  or  that  the  divi- 
dend will  contain  the  divisor  but  1  times. 

Again,  divide  ^  of  a  pear  into  6  equal  parts. 

If  a  whole  pear  were  divided  into  G  equal  parts,  each  part 
would  be  expressed  by  1.  But  since  the  half  of  the  pear 
was  divided,  each  part  will  be  expressed  by  1  of  1,  or  ■^. 

In  the  division  of  fractions  we 'should  note  the  following 
principles : 

1st.  Wlien  the  dividend  is  just  equal  to  the  divisor,  the 
quotient  will  be  1. 

2d.  When  the  dividend  is  greater  than  the  divisor,  the 
quotient  will  be  greater  than  1 . 

Quest. — 138.  W^hat  does  division  of  whole  numbers  explain  ?  In  divisioa 
of  fractious,  may  tlie  divisor  exceed  the  dividend  ?  How  will  the  quotien 
then  compare  witli  1  ?  If  an  apple  be  divided  in  2  equal  parts,  what  will 
express  each  part?  If  half  an  apple  be  divided  into  4  equal  parts,  what 
will  express  one  of  the  parts?  What  is  one -half  of  one-half?  What  is  one- 
sixth  of  one-half?  What  principles  do  you  note  in  the  division  of  fraction*? 
When  will  the  quotient  be  1  ?     When  greater  than  1  ? 


146  DIVISION    OF    VULGAR    FRACTIONS. 

3cl.  When  the  dividend  is  less  than  the  divisor,  the  quo- 
tient will  be  less  than  1 . 

4th.  The  quotient  will  be  just  so  many  times  greater  than 
1,  as  the  dividend  is  greater  than  the  divisor. 

5th.  The  quotient  will  be  just  as  many  times  less  than  1 
as  the  dividend  is  less  than  the  divisor. 

CASE    I. 

139.  To  divide  a  fraction  by  a  whole  number. 

Divide  the  numerator  or  multiply  the  denominator  hy  the 
whole  number, 

EXAMPLES. 

OPERATION. 

4  4  4       2 


3  •  3x2       6^3 

4  ^        2)4       2 
3  3        3 


1.  Divide  I  by  2.    . 
In  the  first  operation  we 

divide  the  fraction  by  mul- 
tiplying the  denominator 
(Art.  100) :  in  the  second 
we  divide  the  numerator 
(Art.  99),  giving  the  same  result  in  both  cases. 

2.  Divide  i|  by  9.  Ans,  

3.  Divide  \Ogp  by  15.  Ans,  

4.  Divide  |^f|  by  19.  Ans,  

5.  Divide  ^j  by  15.  Ans.  

6.  Divide  f^  by  8.  Ans.  

7.  Divide  f^  by  37.  Ans.  

CASE    II. 

140.  To  divide  one  fraction  by  another. 

EXAMPLES. 

1.  Let  it  be  required  to  divide  ^  by  |-. 

The  true  quotient  will  be  expressed  by  the  complex  frac- 

10 

tion  ^. 

8" 

Quest.— -When  will  the  quotient  be  less  than  1  ?  When  greater  than  1 , 
how  many  times  greater?  When  less  than  1,  how  many  times  less?  139 
In  how  many  ways  may  a  fraction  be  divided  by  a  whole  number?  140 
How  do  you  divide  one  fraction  by  another? 


DIVISION    OF    VULGAR    FRACTIONS.  147 

Let  the  terms  of  this  fraction  be  now  multiplied  by  the  de- 
nominator with  its  terms  inverted :  this  will  not  alter  the 
value  of  the  fraction  (Art.  102),  and  we  shall  then  have. 

10  iO.  V    8  10    V    8 

¥  =  ¥#T  =  '-^  =  if  X  f  =  quotient. 

It  will  be  seen  that  the  quotient  is  obtained  by  simply  multi- 
plying the  numerator  by  the  denominator  with  its  terms  in- 
verted. This  quotient  may  be  further  simplified  by  cancelling 
the  common  factors  5  and  8,  giving  f  for  the  true  quotient 

SECOND    METHOD    OF    PROOF. 

OPERATION. 


iO   _:_   K  10 

10     V    8  —     8  0 

T20   '^  o  —  T2or* 


Let  us  first  divide  the  dividend  by 
5.  This  is  done  by  multiplying  the 
denominator  (Art.  100),  which  gives 
^^.  ^ut  the  divisor  being  but  \  of  5, 
this  quotient  is  8  times  too  small,  since  the  eighth  of  a  num- 
ber will  be  contained  in  the  dividend  8  times  more  than  th© 
number  itself.  Therefore,  by  multiplying  -^^  by  8,  we  ob- 
tain -^^  for  the  true  quotient. 

Hence,  to  divide  one  fraction  by  another, 

Reduce  compound  and  complex  fractions  to  simple  ones^  also 
whole  and  mixed  numbers  to  improper  fractions :  then  mul 
tiply  the  dividend  by  the  divisor  with  its  terms  inverted,  and 
the  product  reduced  to  its  simplest  terms  will  be  the  quotient 
sought. 


EXAMPLES. 

1. 

Divide  |  by  \. 

Ans,  

2. 

Divide  3\  by  -J . 

Ans.  

3. 

Divide  16i  of^by  4^.. 

Ans,  

4. 

Divide  44^  by  flf. 

Ans,  

5. 

Divide  371^  by  ^. 

Ans.  - — 

6. 

Divide  ^^  by  ^^. 

Ans.  

7. 

Divide  ^  of  f  by  |  of  f . 

Ans.  

8. 

Divide  5  by  ^. 

Ans.  

9. 

IMvide  52054  by  4  of  91. 

Ans.  

148  DIVISION    OF    VULGAR    FRACTIONS. 

10.  Divide  100  by  4|-.  Ans.  

1 1 .  Divide  f  of  |-  by  f .  Ans.  -^ 

12.  Divide  |  of  50  by  4i.  Ans.  

13.  Divide  14|  of  i  by  3^  of  6.  Ans.  

54J- 

14.  Divide  34i  by  -— |-.  Ans.  

93jj 

15.  Divide  iii  by  ^.  Ans.  

16.  What  number  multiplied  by  J  will  give  15^  for  the 
product  ? 

17.  What  part  of  108  is  ^?  Ans.  

18.  What  number  is  that  which,  if  multiplied  by  -I  of  |^  of 
15^,  will  produce  f  ? 

19.  If  7lb.  of  sugar  cost  ^  of  a  dollar,  what  is  the  price 
per  pound  ? 

20.  If  1^  of  a  dollar  will  pay  for  lO^lb.  of  nails,  how  much 
is  the  price  per  pound  ? 

21.  If  "1^  of  a  yard  of  cloth  cost  $3,  what  is  the  price  pei 
yard? 

22.  If  $21^  will  buy  7^  barrels  of  apples,  how  much  are 
they  per  barrel  ? 

23.  If  4i  gallons  of  molasses  cost  $2|-,  how  much  is  it 
per  quart  ? 

24.  If  l^hhd.  of  wine  cost  $250^,  how  much  is  the  wine 
per  quart  ? 

25.  If  8  pounds  of  tea  cost  7|-  of  a  dollar,  how  much  is  it 
per  pound  ? 

26.  In  8i  weeks  a  family  consumes  165|^  pounds  of  but- 
ter :  how  much  do  they  consume  a  week  ? 

27.  If  a  piece  of  cloth  containing  176-|  yards  costs  $375^, 
what  does  it  cost  per  yard  ? 

28.  Divide  15^  of  -f  of  -J-  of  ^  by  ^  of  ;^  of  i  of  ^. 


DECIMAL    FRACTIONS.  149 


DECIMAL  FRACTIONS. 

• 

141.  If  the  unit  1  be  divided  into  10  equal  parts,  the  parts 
are  called  tenths,  because  each  part  is  one-tenth  of  unity. 

If  the  unit  1  be  divided  into  one  hundred  equal  parts,  the 
pans  are  called  hundredths,  because  each  part  is  one-hun- 
dredth of  unity. 

If  the  unit  1  be  divided  into  one  thousand  equal  parts,  the 
parts  are  called  thousandths,  because  each  part  is  one-thou- 
sandth of  unity :  and  we  have  similar  expressions  for  the 
parts  when  the  unit  is  divided  into  ten  thousand,  one  hundred 
thousand,  &:c.,  equal  parts. 

The   division  of  the   unit  into   tenths,  hundredths, '  thou- 
sandths, &c.,  forms  a   system  of  numbers   called  Decimal 
Fractions.     They  may  be  written, 

Four-tenths,  -         -         -         -         -         -  y^ 

Six-tenths,  ------  ^. 

Forty-five  hundredths,  -----  .j^. 

125  thousandths, i¥ot- 

1047  ten  thousandths, tW^- 

From  which  we  see,  that  in  each  case  the  denominator 
gives  denomination  or  name  to  the  fraction ;  that  is,  deter- 
mines whether  the  parts  are  tenths,  hundredths,  thousandths, 
&c. 

142.  The  denominators  of  decimal  fractions  are  seldom 
set  down.     The  fractions  are  usually  expressed  by  means  of 

Quest. — 141.  When  the  unit  1  is  divided  into  10  equal  parts,  what  is 
each  part  called?  What  is  each  part  called  when  it  is  divided  into  100 
equal  parts?  When  into  1000?  Into  10,000,  &c.?  How  are  decimal 
fractions  formed  ?  What  gives  denomination  to  the  fraction  ?  142.  Are 
the  denominators  of  decimal  fractions  generally  set  down  ?  How  are  the 
firactions  expressed? 


IS  written 

- 

- 

.4 

((        « 

- 

- 

.45 

((        (( 

. 

.   - 

.125 

t(        (( 

- 

- 

.1047. 

&c., 

&c. 

150  DECIMAL    FRACTIONS. 

a  comma,  or  period,  which  is  called  the  decimal  point,  and 
is  placed  at  the  left  of  the  numerator. 
Thus,     ^ 
roo 

1  2_5_ 

roGo 

1  047 

roooo 

&C., 

This  manner  of  expressing  decimal  fractions  is  a  mere 
language,  and  is  used  to  avoid  the  inconvenience  of  writing 
the  denominators.'  The  denominator,  however,  of  every  deci- 
mal fraction  is  always  understood.  It  is  a  unit  1,  with  as 
many  ciphers  annexed  as  there  are  places  of  figures  in  the  nu- 
merator. 

The  place  next  to  the  decimal  point,  is  called  the  place  of 
tenths,  and  its  unit  is  1  tenth.  The  next  place  to  the  right 
is  the  place  of  hundredths,  and  its  unit  is  1  hundredth; 
the  next  is  the  place  of  thousandths,  and  its  unit  is  1  thou- 
sandth ;   and  similarly  for  places  still  to  the  right. 

^   DECIMAL    NUMERATION    TABLE. 


^  Tenths. 
Hundredths. 
Thousandths. 
Tenths  of  thousandt 
Hundredths  of  thousj 
Millionths. 
Tenths  of  millionths 
&c.,             &c. 

is 

read 

4  tenths. 

.6  4 

64  hundredths. 

.0  6  4 

64  thousandths. 

.6  7  5*  4 

6754  ten-thousandths. 

.01234 

1234  hundred-thousandths. 

0  0  7  6  5  4 

7654  milHonths. 

.0043604 

43604  ten-millionths. 

Quest. — Is  the  denominator  understood  ?     What  is  it  ?  What  is  the 

place  next  the  decimal  point  called  ?     What  is  its  unit  ?  What  is  the 

next  place  called?     What  is  its  unit?     The  next?     Its  unit?     Which 
way  are  decimals  numerated  ? 


DECIMAL    FRACTIONS.  151 

Decimal  fractions  are  numerated  from  the  left  hand  to  the 
right,  beginning  with  the  tenths,  hmidredths,  &c.,  as  in  the 
table. 

143.  Let  us  now  write  and  numerate  the  following  deci- 
mals. 

Four-tenths,        -         -         -         -         A, 

Four  hundredths,         -         -         -         .0  4. 

Four  thousandths,        -         -         -         .0  0  4, 

Four  ten-thousandths,  -         -         .0  0  0  4. 

Four  hundred  thousandths,  -         -         .0'  0  0  0  4. 

Four  millionths,  -         -         -         .000004. 

Four  ten-millionths,     -         -         -         .0000004. 

Here  we  see,  that  the  same  figure  expresses  units  of  different 
values,  according  to  the  place  which  it  occupies. 
But  1^  of  To     ^^  equal  to  yuo  —  •^'^• 

To'oo  =  -004., 
To^  =  -0004. 
tWooo  -  -00004. 
"         Too^ooo  =  -000004. 
fo  of    Toowoo         "  "       ro ooWoo  =  .0000004. 

Therefore  the  value  of  the  units  of  the  different  places,  in 
passing  from  the  left  to  the  right,  diminishes  according  to 
the  scale  of  tens. 

Hence,  ten  of  the  parts  in  any  one  of  the  places  are  equal 
to  one  of  the  parts  in  the  place  next  to  the  left ;  that  is,  ten 
thousandths  make  one  hundredth,  ten  hundredths  make  one- 
tenth,  and  ten  tenths  a  unit  1 . 

This  law  of  increase  from  the  right  hand  towards  the  left, 
is  the  same  as  in  whole  numbers.  Therefore,  whole  numbers 
and  decimal  fractions  may  he  united  by  placing  the  decimal 
point  between  them.     Thus, 

Quest. '—Ids.  Does  the  value  of  the  unit  of  a  figure  depend  upon  the 
place  which  it  occupies  ?  How  does  the  value  of  the  unit  change  from 
the  left  towards  the  right  ?  What  do  ten  parts  of  any  one  place  make  ? 
How  do  the  units  of  place  increase  from  the  right  towards  the  left? 
How  may  whole  numbers  be  joined  with  decimals  ? 


I^of 

4 

To 

T^of 

loo 

1^0  of 

looo 

A-  of 

4 

10000 

l^of 

4 

100000 

152 


DECIMAL    FRACTIONS. 
Whole  numbers. 


I"!i|.f5      IIM 


Decimals. 

1 

.   ns 

-     1 

a!. 

'T3     3 

^ 

1    1 

i 

05    1     O 

r3 

i  1  ^    ^       . 

S 

"tS    -T^    «<_,      -*->       CO 

«*« 

.  -S    «   °  -S  :S 

o 

836^0641      .     047897  6. 

A  number  composed  partly  of  a  whole  number  and  partly 
of  a  decimal,  is  called  a  mixed  number. 

Write  the  following  numbers  in  figures,  and  numerate 
them. 

1.  Forty-one,  and  three  tenths.  41.3. 

2.  Sixteen,  and  three  millionths.  16.000003 

3.  Five,  and  nine  hundredths.  5.09, 

4.  Sixty-five,  and  fifteen  thousandths. 

5.  Eighty,  and  three  millionths. 

6.  Two,  and  three  hundred  millionths. 

7.  Four  hundred  and  ninety-two  thousandths. 

8.  Three  thousand,  and  twenty-one  ten-thousandths. 

9.  Forty-seven,  and  twenty-one  ten-thousandths. 

10.  Fifteen  hundred  and  three  millionths. 

11.  Thirty-nine,  and  six  hundred  and  forty  thousandths. 

12.  Three  thousand,  eight  hundred  and  forty  millionths. 

13.  Six  hundred  and  fifty  thousandths. 

14.  Fifty  thousand,  and  four  hundredths. 

15.  Six  hundred,  and  eighteen  ten-thousandths. 

16.  Three  millionths. 

17.  Thirty-nine  hundred- thousandths. 

144.  The  denominations  of  Federal  Money  will  correspond 
to  the  decimal  division,  if  we  regard  1  dollar  as  the  unit. 

Quest. — What  is  a  number  called  when  composed  partly  of  whole  num- 
bers and  partly  of  decimals?  144.  If  the  denominations  of  Federal  Mdiiey 
bo  expressed  decimally,  ifhai  is  the  unit  ? 


DECIMAL    FRACTIONS.  153 

For,  the  dimes  are  tenths  of  the  dollar,  the  cents  are  hun- 
dredths of  the  dollar,  and  the  mills,  being  tenths  of  the  cent, 
are  thousandths  of  the  dollar. 

EXAMPLES. 

1.  Express  $17,  3  dimes  8  cents  and  9  mills  decimally. 

2.  Express  $92,  8  dimes  9  cents  5  mills  decimally. 

3.  Express  $107,  9  dimes  6  cents  8  mills  decimally. 

4.  Express  $47  and  25  cents  decimally. 

5.  Express  $39,  39  cents  and  7  mills  decimally. 

6.  Express  $12  and  3  mills  decimally.  Ans.  

7.  Express  $147  and  4  cents  decimally.         Ans. 

8.  Express  $148,  4  mills  decimally.  Ans. 

9.  Express  four  dollars,  six  mills  decimally.  Ans.  

10.  Express  $14,  3  cents  9  mills  decimally.     Ans.  

11.  Express  $149,  33  cents  2  mills  decimally. 

12.  Express  $1328,  5  mills  decimally.  Ans.  

13.  Express  9  dimes  4  mills  decimally.  Ans.  

14.  Express  5  cents  8  mills  decimally.  Ans.  

15.  Express  $3856,  2  cents  decimally.  Ans.  

145.  A  cipher  is  annexed  to  a  number  when  it  is  placed 
on  the  right  of  it.  If  ciphers  be  annexed  to  the  numerator  of 
a  decimal  fraction,  the  same  number  of  ciphers  must  also  be 
annexed  to  the  denominator ;  for  there  must  always  be  as 
many  ciphers  in  the  denominator  as  there  are  places  of  fig- 
ures in  the  numerator  (Art.  142).  The  numerator  and  de- 
nominator will  therefore  be  multiplied  by  the  same  number, 
and  consequently  the  value  of  the  fraction  will  not  be  changed 
(Art.  102).     Hence, 

Annexing  ciphers  to  a  decimal  fraction  does  not  alter  its 
value. 

Quest. — What  part  of  a  dollar  is  one  dime  ?  What  part  of  a  dime  is  a 
cent  ?  What  part  of  a  cent  is  a  mill  ?  What  part  of  a  dollar  is  1  cent  ? 
1  mill  ?  145.  When  is  a  cipher  annexed  to  a  number  ?  Does  the  annex- 
ing o?  ciphers  to  a  decimal  alter  its  value?  Why  not?  What  do  three- 
tenths  become  by  annexing  a  cipher?     What  by  annexuig  two  ciphers? 


154  DECIMAL    FRACTIONS. 

We  may  take  as  an  example  the  decimal  .3  =:  ^.     If, 
now,  we  annex  a  cipher  to  the  numerator,  we  must,  at  the 
same  time,  annex  one  to  the  denominator,  which  gives 
.30      =       j^  by  annexing  one  cipher, 
.300    =    TU^wo  ^y  annexing  two  ciphers, 
.3000  =z  ^Q%\  all  of  which  are  equal  to  -5^  =  .3. 

Alan        ^  5      -       ^0   —     50     —     500  —    500 

Also,     .0  —  YO    —  '^^   —  TOOT  —    '^^^  —  1000* 

Also,  .8  =:  .80  =  .800  =.  .8000  =:  .80000. 

146.  Prefixing  a  cipher  is  placing  it  on  the  left  of  a  num- 
ber. If  ciphers  be  prefixed  to  the  numerator  of  a  decimal 
fraction,  that  is,  placed  at  the  left  hand  of  the  significant  fig- 
ures, the  same  number  of  ciphers  must  be  annexed  to  the 
denominator.  Now,  the  numerator  will  remain  unchanged 
while  the  denominator  will  be  increased  ten  times  for  every 
cipher  which  is  annexed,  and  the  value  of  the  fraction  will 
be  decreased  in  the  same  proportion  (Art.  100).     Hence, 

Prefixing  ciphers  to  a  decimal  .fraction  diminishes  its  value 
ten  times  for  every  cipher  pi'efxed. 

Take  as  an  example  the  fraction  .2  ==  y^. 

.02       =r       j^  by  prefixing  one  cipher, 
.002    =    xTyVo  ^y  prefixing  two  ciphers, 
.0002  =  TWooQ  ^y  pi'efixing  three  ciphers  : 
in  which  the  fraction  is  diminished  ten  times  for  every  cipher 
prefixed. 

Also,  .03  becomes  .003  by  prefixing '  one  cipher;  and 
.0003  by  prefixing  two. 

Quest. — What  does  .8  become  by  annexing  a  cipher?  By  annexing 
two  ciphers  ?  By  annexing  three  ciphers  ?  146.  When  is  a  cipher  pre- 
fixed to  a  number  ?  When  prefixed  to  a  decimal,  does  it  increase  the  nu- 
merator? Does  it  increase  the  denominator?  What  effect  then  has  it  on 
the  vahie  of  the  fraction  ?  What  does  .5  become  by  prefixing  a  cipher  ?  By 
prefixing  two  ciphers  ?  By  prefixing  three  ?  What  does  .07  become  by 
prefixing  a  cipher  ?  By  prefixing  two  ?  By  prefixing  three  ?  By  prefixing 
four? 


ADDITION    OF    DECIMAL    FRACTIONS.  155 


ADDITION  OF  DECIMAL  FRACTIONS. 

147.  It  must  be  recollected  that  only  like  parts  of  unity 
can  be  added  together,  and  therefore  in  setting  down  the 
numbers  for  addition,  the  figures  occupying  places  of  the 
same  value  must  be  placed  in  the  same  column. 

The  addition  of  decimal  fractions  is  then  made  in  the  same 
manner  as  that  of  whole  numbers. 

Add  37.04,  704.3,  and  .0376  together. 


OPERATION 
37.04 

704.3 
.0376 

741.3776 


In  this  example,  we  place  the  tenths 
under  tenths,  the  hundredths  under  hun- 
dredths, and  this  brings  the  decimal  points 
and  the  like  parts  of  the  unit  in  the  same 
column.  We  then  add  as  in  whole 
numbers. 

Hence,  for  addition  of  decimals, 

I.  Set  down  the  numbers  to  he  added  so  that  tenths  shall  fall 
under  tenths,  hundredths  under  hundredths,  <Sfc.  This  will  bring 
all  the  decimal  points  under  each  other. 

II.  Then  add  as  in  simple  numbers  and  point  off  in  the  sum^ 
from  the  right  hand,  so  many  places  for  decimals  as  are  equal 
to  the  greatest  number  of  places  in  any  of  the  added  numbers. 

EXAMPLES, 

1.  Add  6.035,  763.196,  445.3741,  and  91.5754  together. 

2.  Add  465.103113,  .78012,  1.34976,  .3549,  and  61.11. 

3.  Add  57.406  +  97.004  +  4  -f  .6  +  .06  +  .3. 

4.  Add  .0009  -f  1.0436  +  .4  +  .05  +  .047. 

5.  Add  .0049  +  49.0426  +  37.0410  +  360.0039. 

6.  Add  5.714,  3.456,  .543,  17.4957  together. 


Quest. — 147.  What  parts  of  unity  may  be  added  together?  How  do 
you  set  down  the  numbers  for  addition  ?  How  will  the  decimal  points  fall  '> 
How  do  you  then  add  ?  How  many  decimal  places  do  you  point  off  in 
the  sum? 


156  SUBTRACTION    OF    DECIMAL    FRACTIONS. 

7.  Add  3.754,  47.5,  .00857,  37.5  together. 

8.  Add  54.34,  .375,  14.795,  1.5  together. 

9.  Add  7]  .25,   1.749,  1759.5,  3.1  together. 

10.  Add  375.94,  5.732,  14.375,  1.5  together. 

11.  Add  .005,   .0057,  31.008,  .00594  together. 

12.  Required  the  sum  of  9  tenths,  19  hundredths,  18  thou- 
sandths, 211  hundred-thousandths,  and  19  millionths. 

13.  Required  the  sum  of  twenty -nme  and  3  tenths,  four 
hundred  and  sixty-five,  and  two  hundred  and  twenty-one 
thousandths. 

14.  Required  the  sum  of  two  hundred  dollars  one  dime 
three  cents  and  nine  mills,  four  hundred  and  forty  dollars 
nine  mills,  and  one  dollar  one  dime  and  one  mill. 

15.  What  is  the  sum  of  one  tenth,  one  hundredth,  and  one 
thousandth  ? 

16.  What  is  the  sum  of  4,  and  6  ten-thousandths? 

17.  What  is  the  sum  of  3  thousandths,  9  millionths,  5  hun- 
dredths, 6  hundredths,  3  tenths,  and  2  units  ? 

18.  Required,  in  dollars  and  decimals,  the  sum  of  one  dol- 
lar one  dime  one  cent  one  mill,  six  dollars  three  mills,  four 
dollars  eight  cents,  nine  dollars  six  mills,  one  hundred  dollars 

'six  dimes,  nine  dimes  one  mill,  and  eight  dollars  six  cents. 

19.  What  is  the  sum  of  4  dollars  6  cents,  9  dollars  3  mills, 
14  dollars  3  dimes  9  cents  1  mill,  104  dollars  9  dimes  9 
cents  9  mills,  999  dollars  9  dimes  1  mill,  4  mills,  6  mills,  and 
1  mill  ? 

SUBTRACTION  OF  DECIMAL  FRACTIONS. 

148.   Subtraction  of  Decimal  Fractions  is  the  process  of 
finding  the  difference  between  two  decimal  numbers. 
1.  From  3.275  take  .0879. 
In  this  example  a  cipher  is  annexed -to  operation. 


the  minuend  to  make  the  number  of  deci- 
mal places  equal  to  the  number  in  the 
subtrahend.  This  does  not  alter  the  value 
of  the  minuend  (Art.  145). 


3.2750 

.0879 

3.1871 


SUBTRACTION    OF    DECIMAL    FRACTIONS.  157 

Hence,  for  the  subtraction  of  decimal  numbers, 

1.  Set  down  the  less  nmnher  under  the  greater^  so  that 
figures  occupying  places  of  the  same  value  shall  fall  in  the 
same  column, 

II.  Then  subtract  as  in  simple  numbers,  and  point  off  in  the  re- 
mainder, from  the  right  hand,  as  many  places  for  decimals  as  are 
equal  to  the  greatest  number  of  places  in  either  of  the  given  numbers 

EXAMPLES. 

2.  From  3278  take  .0879.  Ans.  

3.  From  291.10001  take  41.496.  Ans,  

4.  From  10.00001  take  .111111.  Ans.  

5.  Required  the  difference  between  57.49  and  5.768. 

6.  Wh^t  is  the  difference  between  .3054  and  3.075? 

7.  Required  the  difference  between  1745.3  and  173.45. 

8.  What  is  the  difference  between  seven-tenths  and  54 
ten-thousandths  1 

9.  What  is  the  difference  between  .105  and  1.00075? 

10.  What  is  the  difference  between  150.43  and  754.355? 

11.  From  1754.75,4  take  375.49478.  Ans,  

12.  Take  75.304  from  175.01.  Ans,  

13.  Required  the  difference  between  17.541  and  35.49. 

14.  Required  the  difference  between  7  tenths  and  7  mil- 
lion ths. 

15.  From  396  take  8  ten-thousandths.  Ans.  

16.  From  1  take  one-thousandth.  Ans,  

17.  From  6374  take  one-tenth.  Ans,  

18.  From  365.0075  take  5  millionths.  Ans,  

19.  From  21.004  take  98  ten-thousandths.        Ans,  

20.  From  260.3609  take  47  ten-millionths.       Ans.  

21.  From  10.0302  take  19  millionths.  Ans,  

22.  From  2.03  take  6  ten-thousandths  Ans,  

Quest. — 148.  What  does  subtraction  teach?  How  do  you  set  down  the 
numbers  for  subtraction?  How  do  you  then  subtract?  How  many  deci- 
mal places  do  yon  point  off  in  the  remainder  ? 


158  MULTIPLICATION    OF    DECIMAL    FRACTIONS. 


MULTIPLICATION  OF  DECIMAL  FRACTIONS. 

149.— 1.  Multiply  .37  by  .8. 

We  may  first  write    .37  =  j^o'  ^^^^  -^  —  i^* 


OPERATION. 
Q7   37 

Q   8 

.4^JO    —    1000* 

=  .296. 


If,  now,  we  multiply  the  fraction  y^ 
by  3^,  we  find  the  product  to  be  jVcfo  ' 
the  number  of  ciphers  in  the  denomina- 
tor of  this  product  is  equal  to  the  number 
of  decimal  places  in  the  two  factors,  and 
the  same  will  be  true  for  any  two  factors 
whatever. 

2.  Multiply  .3  by  .02. 

OPERATION. 

.3  X  .02  =  t'o  X  T%'o  =  roVo  =  -006  answSr. 

Now,  to  express  the  6  thousandths  decimally,  we  have  to 
prefix  two  ciphers  to  the  6,  and  this  makes  as  many  decimal 
places  in  the  product  as  there  are  in  both  multiplicand  and 
multiplier. 

Therefore,  to  multiply  one  decimal  by  another, 

^  Multiple/  as  in  simple  numbers^  and  point  off  in  the  product, 
from  the  right  handy  as  many  figures  for  decimals  as  are  equal 
to  the  number  of  decimal  places  in  the  multiplicand  and  multi- 
plier ;  and  if  there  be  not  so  many  in  the  product,  supply  tht 
deficiency  by  prefixing  ciphers. 

EXAMPLES. 

1.  Multiply  3.049  by  .012.  Ans.   .036588. 

(2.)  (3.) 

Multiply  365.491  Multiply  496.0135 

by  .901  by  1.496 

Ans.  Ans. 


Quest. — 149.  After  multiplying,  how  many  decimal  ]»laces  will  you  point 
off  in  the  product?  When  there  are  not  so  many  in  the  product,  what  do 
you  do?     Give  the  rule  for  the  multiplication  of  decimals. 


MULTIPLICATION    OF    DECIMAL  FRACTIONS.           159 

4.  Multiply  one  and  one  millionth  by  one  thousandth. 

5.  Multiply  473.54  by  .057.  Ans.  

6.  Multiply  137.549  by  75.437.  Ans.  

7.  Multiply  3.7495  by  73487.  Ans.  

8.  Multiply  .04375  by  .47134.  Ans,  

9.  Multiply  .371343  by  75493.  Ans.  

10.  Multiply  49.0754  by  3.5714.  Ans.  

11.  Multiply  .573005  by  .000754.  Ans.  

12.  Multiply  .375494  by  574.375.  Ans. 


13.  Multiply  two  hundred  and  ninety-four  millionths,  by 
one  millionth. 

14.  Multiply  three  hundred,  and  twenty-seven  hundredths 
by  62.  •  Ans.  

15.  Multiply  93.01401  by  10.03962.  Ans.  

16.  What  is  the  product  of  five-tenths  by  five-tenths  ? 

17.  What  is  the  product  of  five-tenths  by  five  thousandths  ? 

18.  Multiply  596.04  by  0.000012.  Ans.  

19.  Multiply  38049.079  by  0.000016.  Ans.  

20.  Multiply  1192.08  by  0.000024.  Ans.  

21.  Multiply  76098.158  by  0.000032.  Ans.  

CONTRACTION    IN    MULTIPLICATION. 

150.  Contraction  in  the  multiplication  of  decimals  is  a 
short  method  of  finding  the  product  of  two  decimal  numbers 
in  such  a  manner,  that  it  shall  contain  but  a  given  number 
of  decimal  places. 

1.  Let  it  be  required  to  find  the  product  of  2.38645  multi- 
plied by  38.2175,  in  such  a  manner  that  it  shall  contain  but 
four  decimal  places. 

In  this  example  it  is  proposed  to  take  the  multiplicand 
2.38645,  38  times,  then  2  tenths  times,  then  1  hundredth 
times,  then  7  thousandth  times,  then  5  ten-thousandth  times, 

Quest. — 150.  What  is  contraction  in  the  multiplication  of  decimals? 
What  is  proposed  in  the  example  ?  How  are  the  niimbei-s  written  down  for 
multiplication  ? 


160  MULTIPLICATION    OF    DECIMAL    FRACTIONS. 

and  the  sum  of  these  several  products  will  be  the  produc* 
sought. 

Write  the  unit  figure  of  the  multiplier 
directly  under  that  place  of  the  multi- 
plicand which  is  to  be  retained  in  the 
product,  and  the  remaining  places  of  in- 
teger numbers,  if  any,  to  the  right,  and 
then  write  the  decimal  places  to  the  left 
in  their  order,  tenths,  hundredths,  (fee. 


2.38645 
5712.83 


715935 

190916 

4773 

239 

167 

12 

91.2042 


When  the  numbers  are  so  written,  the 
product  of  any  figure  in  the  multiplier  by 
the  figure  of  the  multiplicand  directly  over  it,  will  he  of  the 
same  order  of  value  as  the  last  figure  to  be'  retained  in  th0 
product.  Therefore,  the  first  figure  of  each  product  is  always 
to  be  arranged  directly  under  the  last  retained  figure  of  the 
multiplicand.  But  it  is  the  whole  of  the  multiplicand  which 
should  be  multiplied  by  each  figure  of  the  multiplier,  and  not 
a  part  of  it  only.  Hence,  to  compensate  for  the  part  omitted, 
we  begin  with  the  figure  to  the  right  of  the  one  directly  over 
any  multiplier,  and  carry  one  when  the  product  is  greater 
than  5  and  less  than  15,  2  when  it  falls  between  15  and  25, 
3  when  it  falls  between  25  and  35,  and  so  on  for  the  higher 
numbers. 

For  example,  when  we  multiply  by  the  8,  instead  of  say- 
ing 8  times  4  are  32,  and  writing  down  the  2,  we  say  first, 
8  times  5  are  40,  and  then  carry  4  to  the  product  32,  which 
gives  36.  So,  when  we  multiply  by  the  last  figure  5,  we  first 
say,  5  times  3  are  15,  then  5  times  2  are  10  and  2  to  carry 
make  12,  which  is  written  down. 

EXAMPLES. 

1.  Multiply  36.74637  by  127.0463,  retaining  three  decimal 
places  in  the  product. 

Quest. — When  the  numbers  are  so  written,  what  will  be  the  order  of 
value  of  the  product  of  any  figure  of  the  multiplier  by  the  figure  directly 
over  it  ?  Where  then  is  the  first  figure  by  each  product  to  be  written  I 
How  do  you  compensate  for  the  part  omitted  ? 


MULTIPLIGATION    OF    DECIMAL    FRACTIONS. 


16h 


CONTRACTION. 

36.74637 
3640.721 

3674637 

734927 

257224 

1470 

220 

11 

4668.489 


COMMON    WAY. 

36.74637 
]  27.0463 


11023911 
22047822 
14698548 
25722459 
7349274 
3674637 
4668.490346931 


2.  Multiply  54.7494367  by  4.714753,  reserving  ^ve  places 
of  decimals  in  the  product.  , 

3.  Multiply  475.710564  by  .3416494,  retaining  three  deci- 
mal places  in  the  product. 

4.  Multiply  3754.4078  by  .734576,  retaining  five  decimal 
places  in  the  product. 

5.  Multiply  4745.679  by  751.4549,  and  reserve  only  whole 
numbers  in  the  product. 

151.  Note. — When  a  decimal  number  is  to  be  multiplied  by  10, 
100,  1000,  &c.,  the  multiplication  may  be  made  by  removing  the 
decimal  point  as  many  places  to  the  right  hand  as  there  are  ciphers 
in  the  multiplier ;  and  if  there  be  not  so  many  figures  on  the  right 
of  the  decimal  point,  supply  the  deficiency  by  annexing  ciphers. 


Thus,  6.79  multiplied  by  < 


Also,  370.036  multiplied  by 


10 

f    67.9 

100 

679. 

1000 

r  ~  ^ 

6790. 

10000 

67900. 

100000 

L  679000. 

10    1 

r    3700.36 

100 

37003.6 

1000 

'^  ~  1 

370036. 

10000 

3700360. 

100000 

37003600. 

Quest. — 151.  How  do  you  multiply  a  decimal  number  by  10,  100, 1000, 
6lc.  ?  If  there  are  not  as  many  decimal  figures  as  there  are  ciphers  in  the 
multiplier,  what  do  you  do  *? 


162  DIVISION    OF    DECIMAL    FRACTIONS. 


DIVISION  OF  DECIMAL  FRACTIONS. 

152.  Division  of  Decimal  Fractions  is  similar  to  tliat  of 
simple  numbers. 

We  have  just  seen  that,  if  one  decimal  fraction  be  multi- 
plied by  another,  the  product  will  contain  as  many  places  of 
decimals  as  there  were  in  both  the  factors.  Now,  if  this 
product  be  divided  by  one  of  the  factors,  the  quotient  will  be 
the  other  factor  (Art.  79)-  Hence,  in  division,  the  dividend 
must  contain  just  as  many  decimal  places  as  the  divisor  and 
quotient  together.  The  quotient^  therefore,  will  contain  as  many 
places  as  the  dividend^  less  the  number  in  the  divisor. 

EXAMPLES, 

1.  Divide  1.38483  by  60.21. 


There  are  five  decimal  places  in 


OPERATION. 


60.21)1.38483(23 
1.2042 


18063 

18063 

A71S.   .023. 


the  dividend,  and  two  in  the  divi- 
sor :  there  must  therefore  be  three 
places  in  the  quotient :  hence  one 
0  must  be  prefixed  to  the  23,  and 
the  decimal  point  placed  before  it. 
Hence,  for  the  division  of  decimals, 

Divide  as  in  simple  numbers,  and  point  off  in  the  quotient, 
from  the  right  hand,  so  many  places  for  decimals  as  the  deci- 
mahplaces  in  the  dividend  exceed  those  in  the  divisor ;  and  if 
there  are  not  so  many,  supply  the  deficiency  by  prefixing  ciphers. 

2.  Divide  4.6842  by  2.11.  Ans.  

3.  Divide  12.82561  by  1.505.  Ans.  

4.  Divide  33.66431  by  1.01.  Ans.  


Quest. — 152.  If  one  decimal  fraction  be  multiplied  by  another,  how  many 
decimal  places  will  there  be  in  the  product  ?  How  does  the  number  of 
decimal  places  in  the  dividend  compare  with  those  in  the  divisor  and  quo- 
tient 1  How  do  you  determine  the  number  of  decimal  places  in  the  quotient  ? 
If  the  divisor  contains  four  places  and  the  dividend  six,  how  many  in  the 
quotient  ?  If  the  divisor  contains  three  places  and  the  dividend  five,  how 
many  in  the  quotient '?     Give  the  rule  for  the  division  of  decimals. 


DIVISION    OF    DECIMAL    FRACTIONS. 


163 


6.  Divide  .010001  by  .01. 
6.  Divide  24.8410  by  .002. 


Ans, 
Ans, 


7.  What  is  the  quotient  of  75.15204,  divided  by  3?  By  .3  ? 
By  .03  ?     By  .003  ?     By  ^.0003  ? 

.  8.  What  is  the  quotient  of  389.27688,  divided  by  8  ?     By 
.08  ?     By  .008  1     By  .0008  ?     By  .00008  ? 

9.  What  is  the  quotient  of  374.598,  divided  by  9  ?  By  .9  ? 
By  .09?     By  .009?     By  .0009  ?     By  .00009  ? 

10.  What  is  the  quotient  of  1528.4086488,  divided  by  6? 
By  .06  ?    By  .006  ?    By  .0006  ?    By  .00006  ?    By  .000006  ? 

11.  Divide  17.543275  by  125.7.  Ans,  

12.  Divide  1437.5435  by  .7493.  Ans,  

13.  Divide  .000177089  by  .0374.  Ans,  

14.  Divide  1674.35520  by  960.  Ans,  

15.  Divide  120463.2000  by  1728.  Ans,  ' 

16.  Divide  47.54936  by  34.75.  Ans,  

17.  Divide  74.35716  by  .00573.  Ans,  

18.  Divide  .37545987  by  75.714.  Ans.  

153.  Note  I. — When  any  decimal  number  is  to  be  divided  by 
10,  100,  1000,  &c.,  the  division  is  made  by  removing  the  decilnal 
point  as  many  places  to  the  left  as  there  are  O's  in  the  divisor ;  and 
if  there  be  not  so  many  figures  on  the  left  of  the  decimal  point,  the 
deficiency  must  be  supplied  by  prefixing  ciphers. 


27.69  divided  by 


642.89  divided  by 


10  -] 

r 

2.769 

100 

.2769 

1000  f 

— 

.02769 

10000 

.002769 

10 

r  64.289 

100 

6.4289 

1000 

>■==.< 

.64289 

10000 

.064289 

100000 

.0064289 

Quest. — 153.  How  do  you  divide  a  decimal  number  by  10,  100,  1000, 
&c.  ?  If  there  be  not  as  many  figures  to  the  left  of  the  decimal  point  as 
ther<*  are  ciphers  in  the  divisor,  what  do  you  do? 


164 


DfVistON    OP    l!)EClMAL    FRACTIONS. 


154.  Note  2. — When  there  are  more  decimal  places  in  the  divi- 
sor than  in  the  dividend,  annex  as  many  ciphers  to  the  dividend  as 
are  necessary  to  make  its  decimal  places  equal  to  those  of  the 
divisor ;  all  the  figures  of  the  quotient  will  then  be  whole  numbers. 
Always  bear  in  mind  that  the  quotient  is.  as  many  times  greater  than 
unity,  as  the  dividend  is  greater  than  the  divisor. 


EXAMPLES. 

I.  Divide  4397.4  by  3.49. 

We  annex  one  0  to  the  dividend. 
Had  it  contained  no  decimal  place 
we  should  have  annexed  two. 


OPERATION. 

3.49)4397.40(1260 
349 
~907 

698 

2094 

2094 


Ans.   1260. 

2.  Divide  1097.01097  by  .100001.  Ans.  

3.  Divide  9811.0047  by  .1629735.  Ans.  

•   4.   Divide  .1  by  .0001.  Ans.  — — 

5.  Divide  10  by  .1.  Ans. 

6.  Divide  6  by  .6.  By  .06.  By  .006.  By  .2.  By  .3. 
By  .003.     By  .5.     By  .005.     By  .000012. 

155.  Note  3. — ^When  it  is  necessary  to  continue  the  division 
farther  than  the  figures  of  the  dividend  will  allow,  we  may  annex 
ciphers  to  it,  and  consider  them  as  decimal  places. 

EXAMPLES. 

1.  Divide  4.25  by  1.25. 

In  this  example,  after  having  ex- 
hausted the  decimals  of  the  dividend, 
we  annex  an  0,  and  then  the  decimal 
places  used  in  the  dividend  will  exceed 
those  in  the  divisor  by  1. 


OPERATION. 

1.25)4.25(3.4 
3.75 
500 
500 
Ans.     O 


Quest. — 154.  If  there  are  more  decimal  places  in  the  divisor  than  in  the 
dividend,  what  do  you  do  ?  What  will  the  figures  of  the  quotient  then  be  ? 
155.  How  do  you  continue  the  division  after  you  have  brought  down  all  th« 
figures  of  the  dividend  ? 


DIVISION    OF    DECIMAL   FRACTIONS. 


165 


2.  Divide  .2  by  .06. 

We  see  in  this  example  that  the  di- 
vision will  never  terminate.  In  such 
cases  the  division  should  be  carried  to 
the  third  or  fourth  place,  which  will 
give  the  answer  true  enough  for  all 
practical  purposes,  and  the  sign  + 
should  then  be  written,  to  show  that 
the  division  may  still  be  continued. 

3.  Divide  37.4  by  4.5. 

4.  Divide  586.4  by  375. 

5.  Divide  94.0369  by  81.032. 


OPERATION. 

.06).20(3.333  + 
18 
20 
18 
~20 
18 
20 
Ans.  3.333  +  . 


Ans. 
Ans. 
Ans. 


REMARKS. 

156.  The  unit  of  Federal  Money,  the  currency  of  the 
United  States,  is  ohe  dollar,  and  all  the  lower  denominations, 
dimes,  cents,  and  mills,  are  decimals  of  the  dollar.  Hence, 
all  the  operations  upon  Federal  Money  are  tlie  same  as  the 
corresponding  operations  on  decimal  fractions. 


APPLICATIONS    IN    THE    FOUR    PRECEDING    RULES. 

1.  A  merchant  sold  4  parcels  of  cloth;  the  1st  contained 
239  and  3  thousandths  yards  ;  the  2d,  6  and  5  tenths  yards ; 
the  3d,  4  and  one  hundredth  yards ;  the  4th,  90  and  one  mil- 
lionth yards  :  how  many  yards  did  he  sell  in  all  ? 

2.  A  merchant  buys  three  chests  of  tea ;  the  first  contains 
70  and  one  thousandth  lb.  ;  the  second,  49  and  one  ten- thou- 
sandth Ih.',  the  third,  36  and  one-tenth  Ih.:  how  much  did 
he  buy  in  all  ? 

3.  What  is  the  sum  of  $20  and  three  hundredths ;  $44 
and  one-tenth,  $6  and  one  thousandth,  and  $18  and  one  hun- 
dredth ? 

Quest. — When  the  division  does  not  terminate,  what  sign  do  you  place 
after  the  quotient?  What  does  it  show?  156.  What  is  the  unit  of  the  cur- 
rency of  the  United  States  ?  What  parts  of  this  unit  ar«  the  inferior  de- 
nominations, dimes,  cents,  and  mills  ? 


166  DIVISION    OF    DECIMAL    FRACTIONS. 

4.  A  puis  in  trade  $1504.342;  B  puts  in  $350,1965; 
C  puts  in  $100.11;  D  puts  in  $99,334;  and  E  puts  ia 
$9001.31  :  what  is  the  whole  amount  put  in  ? 

5.  B  has  $936,  and  A  has  $5,  3  dimes,  and  1  mill :  how 
much  more  money  has  B  than  A 1 

6.  A  merchant  buys  1 12.5  yards  of  cloth,  at  one  dollar  twen- 
ty-five cents  per  yard :  how  much  does  the  whole  come  to  ? 

7.  A  farmer  sells  to  a  merchant  13.12  cords  of  wood  at 
$4.25  per  cord,  and  17  bushels  of  wheat  at  $1.06  per  bushel : 
he  is  to  take  in  payment  13  yards  of  broadcloth  at  $4.07  per 
yard,  and  the  remainder  in  cash :  how  much  money  did  he 
receive  1 

8.  If  1 1  men  had  each  $339  1  dime  9  cents  and  3  mills, 
what  would  be  the  total  amount  of  their  money  ? 

9.  If  one  man  can  remove  5.91  cubic  yards  of  earth  in  a 
day,  how  much  could  38  men  remove  1 

10.  What  is,the  cost  of  24.9  yards  of  cloth,  at  $5.47  per 
yard? 

11.  If  a  man  earns  one  dollar  and  one  mill  per  day,  how 
much  will  he  earn  in  a  year  ? 

12.  What  will  be  the  cost  of  675  thousandths  of  a  cord  of 
wood,  at  $2  per  cord  ? 

13.  A  farmer  purchased  a  farm  containing  56  acres  of 
woodland,  for  which  he  paid  $46,347  per  acre;  176  acres 
of  meadow  land  at  the  rate  of  $59,465  per  acre ;  besides 
which  there  was  a  swamp  on  the  farm  that  covered  37  acres, 
for  which  he  was  charged  $13,836  per  acre.  What  was  the 
area  of  the  land ;  what  its  cost ;  and  what  the  average  price 
per  acre  ? 

14.  A  person  dying  has  $8345  in  cash,  and  6  houses 
valued  at  $4379.837  each;  he  ordered  his  debts  to  be  paid, 
amounting  to  $3976.480,  and  $120  to  be  expended  at  his 
funeral ;  the  residue  was  to  be  divided  among  his  five  sons 
in  the  following  manner :  the  eldest  was  to  have  a  fourth 
part,  and  each  of  the  other  sons  to  have  equal  shares.  What 
was  the  share  of  each  son  ? 


DIVISION    OF    DECIMAL    FRACTIONS.  167 


CONTRACTION    IN    DIVISION. 

157.  Contraction  in  division  is  a  short  method  of  obtaining 
the  quotient  of  one  decimal  number  divided  by  another. 

EXAMPLES 

1.  Divide  754.347385  by  61.34775,  and  let  thB  quotient 
contain  three  places  of  decimals. 

COMMON  METHOD. 

61.34775)754.34738500(12.296        contracted  method. 

61.34775)754.347385(12.296 
61348 

14086 
12269 


61347 

75 

14086 
12269 

988 
550 

1817 
1226 

4385 
9550 

590 
552 

48350 
12975 

38 
36 

353750 
808650 

1 

545100 

1817 
1227 


590 

552 

38 

37 

1 


It  is  plain  that  all  the  work  by  the  common  method,  which 
stands  on  the  right  of  the  vertical  line,  does  not  affect  the 
quotient  figures.  On  what  principle  is  the  work  omitted  in 
the  contracted  method  ? 

In  every  division,  the  first  figure  of  the.  quotient  will  always 
he  of  the  same  order  of  value  as  that  figure  of  the  dividend  un- 
der which  is  written  the  product  of  the  first  figure  of  the  quo- 
tient  by  the  unites  figure  of  the  divisor. 

Having  determined  the  order  of  value  of  the  first  quotient 
figure,  make  use  of  as  many  figures  of  the  divisor  as  you  wish 
places  of  figures  in  the  quotient. 

Let  each  remainder  be  a  new  dividend,  and  in  each  follow- 
ing division  omit  one  figure  to  the  right  hand  of  the  divisor, 

Quest — 157.  What  is  contraction  in  division?  In  every  division,  what 
will  be  the  order  of  the  first  quotient  figure  ?  How  many  figures  of  the 
divisor  will  you  use  ?     How  will  you  then  make  the  division  ? 


168       REDUCTION  OF  DECIMAL  FRACTIONS. 

observing  to  carry  for  the  increase  of  the  figures  cut  oiF,  as  in 
contraction  of  multiplication. 

In  the  example  above,  the  order  of  the  first  quotient  figure 
was  obviously  tens  ;  hence,  as  there  were  three  decimal 
places  required  in  the  quotient,  five  figures  of  the  divisor  must 
be  used. 

2.  Divide  59  by  .74571345,  and  let  the  quotient  contain 
four  places  of  decimals. 

3.  Divide  17493.407704962  by  495.783269,  and  let  the 
quotie'nt  contain  four  places  of  decimals. 

4.  Divide  98.187437  by  8.4765618,  and  let  the  quotient 
contain  ten  places  of  decimals. 

5.  Divide  47194.379457  by  14.73495,  and  let  the  quotient 
contain  as  many  decimal  places  as  there  will  be  integers  in  it. 

REDUCTION  OF  VULGAR  FRACTIONS  TO  DECIMALS. 

158.  The  value  of  every  vulgar  fraction  is  equal  to  the 
quotient  arising  from  dividing  the  numerator  by  the  denomi- 
nator (Art.  94). 

EXAMPLES. 

1 .  What  is  the  value  in  decimals  of  |-  ? 
We  first  divide  9  by  2,  which 

gives  a  quotient  4,  and  1  for  a  re- 
mainder.    Now,  1   is  equal  to  10 
tenths.     If,  then,  we  add  a  cipher, 
2  will  divide  10,  giving  the  quotient  5  tenths.     Hence,  the 
true  quotient  is  4.5. 

2.  What  is  the  value  of  ^^  ? 


OPERATION. 

|=:4i;but      . 
4i  =  4V^  =4.5. 


OPERATION. 

L3  =  3i ;  but 

3i  -  3100  _  3.25. 


We  first  divide  by  4,  which  gives 
a  quotient  3  and  a  remainder  1. 
But  1  is  equal  to  100  hundredths. 
If,  then,  we  add  two  ciphers,  4  will 
divide  the  100,  giving  a  quotient  of  25  hundredths. 

Quest. — What  is  the  order  of  the  first  quotient  figure  in  Ex.  2  ?  In  3  ? 
In  4?  158.  What  is  the  value  of  a  fraction  equal  to?  What  is  the 
value  of  four-halves  ? 


REDUCTION  OF  DECIMAL  FRACTIONS.        169 

Hence,  to  reduce  a  vulgar  fraction  to  a  decimal, 

I.  Annex  one  or  more  ciphers  to  the  numerator  and  then  di' 
vide  by  the  denominator. 

II.  If  there  is  a  remainder,  annex  a  cipher  or  ciphers,  and 
divide  again;  and  continue  to  annex  ciphers  and  to  divide  until 
there  is  no  remainder,  or  until  the -quotient  is  sufficiently  exact: 
the  number  of  decimal  places  to  be  pointed  off  in  the  quotient  is 
the  same  as  the  number  of  ciphers  used ;  and  when  there  are  not 
so  many,  ciphers  must  be  prefixed  to  supply  the  deficiency. 


EXAMPLES. 

1.  Reduce  ^ff  to  its  equivalent  decimal. 

We  here  use  two  ciphers,  and  there- 
fore point  off  two  decimal  places  in  the 
quotient. 


OPERATION. 

125)635(5.08 
625 


1000 
1000 


2.  Reduce  \  and  xrlg  ^^  decimals.  Ans.  — 

3.  Reduce  ^,  f^,  y^o,  and  ^^Vo^  ^^  decimals. 

4.  Reduce  1  and  yyVs  ^^  decimals.  Ans.  — 

5.  Reduce  fx^fliff  ^^  a  decimal.  Ans.  — 

6.  Reduce  f ,  |f  Jf ,  32^,  f74  ^^  decimals.    Ans.  - 

7.  Reduce  jIto  ^^  deciaials.  Ans.  — 


8. 

Reduce  ^Vo  ^^  decimals. 

Ans.  

9. 

Reduce  jFaoo  ^^  decimals. 

Ans,  

10. 

Reduce  ys^^s  ^^  decimals. 

Ans.  

11. 

Reduce  ^  o  4  s  o  o  o  ^^  decimals. 

Ans,  

12. 

Reduce  f  to  decimals. 

Ans.  

13. 

Reduce  ^  to  decimals. 

Ans.  

14. 

Reduce  ^  to  decimals. 

Ans,  

15. 

Reduce  ]-^  to  decimals. 

Ans.  

16. 

Reduce  yfIt  ^^  decimals. 

Ans.  

17. 

Reduce  ^^j  to  decimals. 

Ans.  

Quest. — What  is  the  decimal  value  of  one-half?  Of  three -fourths?  Of 
gix-fourths?  Of  nhie-halves?  Of  seven-halves?  Of  five -fourths?  Of  one- 
fourth?     Give  the  rule  for  redncinjr  a  vulgar  fraction  to  a  decimal 

R 


170  REDUCTION    OF    BENOMlNATE    DECIMALS. 

18.  What  is  the  decimal  value  of  f  of  f  multiplied  by  ^^^2  * 

19.  What  is  the  value  in  decimals  of  J  of  f  of  |-  divided 
byfoff? 

20.  A  man  owns  ■§-  of  a  ship  ;  he  sells  -^^  of  his  share  • 
what  part  is  that  of  the  whole,  expressed  in  decimals  ? 

21.  Bought  W  of  87j-\  bushels  of  wheat  for  ^^  of  7  dollars 
a  bushel :   how  much  did  it  come  to,  expressed  in  decimals  ? 

22.  If  a  man  receives  f  of  a  dollar  at  one  time,  1\  at 
another,  and  8^  at  a  third :  how  many  in  all,  expressed  in 
decimals  ? 

23.  What  decimal  is  equal  to  f  of  18,  and  j^j  of  ^  of  1^^ 
added  together  ? 

24.  What  decimal  is  equal  to  f  of  6  taken  from  |  of  8f  ? 

25.  What  decimal  is  equal  to  W^  \pf  f,  added  together^ 

REDUCTION  OF  DENOMINATE  DECIMALS. 

159.  We  have  seen  that  a  denominate  number  is  one  in 
which  the  kind  of  unit  is  denominated  or  expressed  (Art.  14). 

A  denominate  decimal  is  a  decimal  fraction  in  which  the 
kind  of  unit  that  has  been  divided  is  expressed.  Thus,  .5 
of  a  £,  and  .6  of  a  shilling  are  denominate  decimals  :  the 
unit  that  was  divided  in  the  first  fraction  being  £1,  and  that 
in  the  second  1  shilling, 

CASE    I. 

160.  To  find  the  value  of  a  denominate  number  in  deci- 
mals of  a  higher  denomination. 

1.  Reduce  9d.  to  the  decimal  of  a  £. 


We  first  find  that  there  are  240 
pence  in  £l.  We  then  divide  9d.  by 
240,  which  gives  the  quotient  .0375 
of  a  £.  This  is  the  true  value  of  9d. 
in  the  decimal  of  a  £. 


.OPERATION. 

2i0d,=£l 
240)9(.0375 
Ans.  £.0375. 


Quest. — 159.  What  is  a  denomiuate  number?  What  is  a  denomiuate 
decimal?  In  the  decimal  five-tenths  of  a  £,  what  is  the  unit?  In  the 
decimal  six-tenths  of  a  shilling,  what  is  the  unit  ? 


REDUCTION    OF    DENOMINATE    DECIMALS.  171 

Hence,  to  make  the  reduction, 

I.  Consider  how  many  units  of  the  given  denomination  make 
one  unit  of  the  denoinination  to  which  you  would  reduce. 

II.  Divide  the  given  denominate  number  by  the  number  so 
founds  and  the  quotient  will  be  the  value  in  the  required  de- 
nomination. 

EXAMPLES. 

1.  Reduce  14  drams  to  the  decimal  of  a  lb.  avoirdupois. 

2.  Reduce  78^.  to  the  decimal  of  a  JG. 

3.  Reduce  .056  poles  to  the  decimal  of  an  acre. 
^  4.  Reduce  42  minutes  to  the  decimal  of  a  day. 

5.  Reduce  63  pints  to  the  decimal  of  a  peck. 

6.  Reduce  9  hours  to  the  decimal  of  a  day. 

7.  Reduce  375678  feet  to  the  decimal  of  a  mile. 

8.  Reduce  72  yards  to  the  decimal  of  a  rod. 

9.  Reduce  .5  quarts  to  the  decimal  of  a  barrel. 

10.  Reduce  Aft.  Gin.  to  the  decimal  of  a  yard. 

11.  Reduce  7oz.  I9pwt.  of  silver  to  the  decimal  of  a  pound. 

12.  Reduce  9^  months  to  the  decimal  of  a  year. 

13.  Reduce  62  days  to  the  decimal  of  a  year  of  365^  days. 

14.  Reduce  £25  19^.  6^d.  to  the  decimal  of  a  pound. 

15.  Reduce  3qr,  2Ub.  to  the  decimal  of  a  cwt, 

16.  Reduce  5fur.  36rd.  2yd.  2ft.  9in.  to  the  decimal  of  a 
mile. 

17.  Reduce  Acwt.  2^qr.  to  the  decimal  of  a  ton. 

18.  Reduce  3cwt.  lib,  8oz,  to  the  decimal  of  a  ton. 

19.  Reduce  17 hr,  6m.  43sec.  to  the  decimal  of  a  day. 


161.  To  reduce  denominate  numbers  of  different  denomi- 
nations to  an  equivalent  decimal  of  a  given  denomination. 

Quest. — 160.  How  do  you  find  the  value  of  a  denominate  number  in  a 
decimal  of  a  higher  denomination  ? 


OPERATION. 

^d.  =  .75d.  ;  hence, 
9f  ^.  =  9.75d. 
I2)9.75d. 

.81255.,  and 
20)4.81255. 

£.240625 ;  therefore, 
£1  4s.  9^d,  =£1.240625. 


173  REDUCTION    OF    DENOMINATE    DECIMALS. 

1.  Reduce  £1  4^.  9^d.  to  the  denomination  of  pounds. 

We  first  reduce  3  farthings 
to  the  decimal  of  a  penny,  by 
dividing  by  4.  We  then  annex 
the  quotient  .75  to  the  9  pence. 
We  next  divide  by  12,  giving 
.8125,  which  is  the  decimal 
of  a  shilling.  This  we  annex 
to  the  shillings,  and  then  di- 
vide by  20. 

Hence,  to  make  the  reduction, 

Divide  the  lowest  denomination  named,  by  that  number  which 
*  makes  one  of  the  denomination  next  higher,  annexing  ciphers  if 
necessary ;  then  annex  this  quotient  to  the  next  higher  denomi- 
nation, and  divide  as  before  :  proceed  in  the  same  manner  through 
all  the  denominations  to  the  last :  the  last  result  will  be  the 
answer  sought. 

EXAMPLES. 

1.  Reduce  £19  17 s.  3^d.  to  the  decimal  of  a  £. 

2.  Reduce  46^.  6d.  to  the  denomination  of  pounds. 

3.  Reduce  7^d.  to  the  decimal  of  a  shilling. 

4.  Reduce  2lb.  boz.   I2pwt.  I6gr.  troy  to  the  decimal  of 

Silb. 

5.  Reduce  7  feet  6  inches  to  the  denomination  of  yards. 

6.  Reduce  lib.  I2dr.  avoirdupois  to  the  denomination  of 
pounds. 

7.  Reduce  10  leagues  4  furlongs  to  the  denomination  of 
leagues. 

8.  Reduce  7^.  5^d.  to  the  decimal  of  a  pound. 

9.  What  decimal  part  of  a  pound  is  three  halfpence  ? 

10.  Reduce  4^.  7^d.  to  the  decimal  of  a  pound. 

11.  Reduce  loz.  llpwt.  3gr.  to  the  decimal  of  a  pound 
troy. 

Quest. — 161.  How  do  you  reduce  denominate  numbers  of  different  de- 
nominations to  equivalent  decimals  of  a  given  denomination  ? 


REDUCTION    OF    DENOMINATE    DECIMALS.  173 

12.  Reduce  24  grains  to  the  decimal  of  an  ounce  troy. 

13.  Reduce  5oz,  Adr,  avoirdupois  to  the  decimal  of  a  pound 
troy. 

14.  Reduce  3cwt.  Iqr.  14lb.  to  the  decimal  of  a  ton. 

15.  Reduce  2qr.  Iblb.  to  the  decimal  of  a  hundred-weight. 

16.  Reduce  5lb.  lOoz,  Spwt.  I3gr,  troy  to  the  decimal  of  a 
hundred- weight  avoirdupois. 

17.  Reduce  Iqr,  \na.  to  the  decimal  of  a  yard. 

18.  Reduce  2qr.  3na.  to  the  decimal  of  an  English  ell. 

19.  Reduce  27/ds.  2ft.  Q^in.  to  the  decimal  of  a  mile. 

20.  What  decimal  part  of  an  acre  is  IR.  37 P  ? 

21.  What  decimal  part  of  a  hogshead  of  wine  is  2  quarts 
1  pint? 

22.  Reduce  3  bushels  3  pecks  to  the  decimal  of  a  chaldron 
of  36  bushels. 

23.  What  decimal  part  of  a  year  is  3wk.  6da.  7Ar.,  reckon- 
ing 365da.  6hr.  a  year  ? 

24.  Reduce  2.45  shillings  to  the  decimal  of  a  £. 

25.  Reduce  1.047  roods  to  the  decimal  of  an  acre. 

26.  Reduce  176.9  yards  to  the  decimal  of  a  mile. 

CASE    III. 

162.  To  find  the  value  of  a  denominate  decimal  in  terms 
of  integers  of  inferior  denominations. 

1.  What  is  the  value  of  .832296  of  a  £  ? 

We  first  multiply  the  decimal  by 
20,  which  brings  it  to  shillings,  and 
after  cutting  off  from  the  right  as 
many  places  for  decimals  as  in  the 
given  number,  we  have  16^.  and 
the  decimal  .645920  over.  This 
we  reduce  to  pence  by  multiplying 
by  12,  and  then  reduce  to  farthings 
by  multiplying  by  4. 


OPERATION. 

.832296 

20 

16.645920 

12 

7.751040 

4 

3.004160 
Ans,    I6s.  7d.  3far, 


174  REDUCTION    OF    DENOMINATE    DECIMALS. 

Hence,  to  make  the  reduction, 

I.  Consider  how  many  in  the  next  less  denomination  make 
one  of  the  given  denomination,  and  multiply  the  decimal  by  this 
number.  Then  cut  off  from  the  right  hand  as  many  places  as 
there  are  in  the  given  decimal. 

II.  Multiply  the  figures  so  cut  off  by  the  number  which  it 
takes  in  the  next  less  denomination  to  make  one  of  a  higher,  and 
cut  off  as  before.  Proceed  in  the  same  way  to  the  lowest  de- 
nomination:  the  figures  to  the  left  will  form  the  answer  sought 

EXAMPLES. 

1 .  What  is  the  value  of  .625  of  a  ciot.  ?  Ans.  

2.  What  is  the  value  of  .625  of  a  gallon  1         Ans.  

3.  What  is  the  value  of  .004168/5.  troy?  Ans.  • 

4.  What  is  the  value  of  .375  hogshead  of  beer  ? 

5.  What  is  the  value  of  .375  of  a  year  of  365  days  ? 

6.  What  is  the  value  of  .085  of  a  £  ?  Ans.  

7.  What  is  the  value  of  .258  of  a  cwt.  ?  Ans.  


8.  What  is  the  difference  between  .82  of  a  day  and  .64  of 
an  hour  ? 

9.  What  is  the  value  of  2.078  miles  ?  Ans.  

10.  What  is  the  value  of  £.3375  1  Ans.  

11.  What  is  the  value  of  .3375  of  a  ton  ?  Ans.  

12.  What  is  the  value  of  .05  of  an  acre  ?  Ans.  

13.  What  is  the  value  of  .875  pipes  of  wine  ? 

14.  What  is  the  value  of  .046875  of  a  pound,  avoirdupois? 

15.  What  is  the  value  of  .56986  of  a  year  of  305 J  days  ? 

16.  What  is  the  value  of  £2.092  ?  Ans.  

17.  What  is  the  value  of  £5.64  ?  Ans.  

18.  What  is  the  value  of  .36974  of  a  last,  wool  wejght  ? 

19.  What  is  the  value  of  .827364^/-.,  corn  measure  ? 

20.  What  is  the  value  of  .093765Sj.  ?  Ans. 


Quest. — 162.  How  do  you  find  the  value  of  a  denominate  decimal  in  in 
tegers  of  inferior  denominations?  What  is  the  value  in  shillings  of  cue- 
half  of  a  £?     In  pence  of  one-half  of  a  shilling? 


CIRCULATING    OR    REPEATING    DECIMALS.  175 


OPERATION. 
12)50000 

.4166  + 


CIRCULATING  OR  REPEATING  DECIMALS. 

163.  We  have  seen  that  in  changing  a  vulgar  into  a  deci- 
mal fraction,  cases  will  arise  in  which  the  division  does  not 
terminate,  and  then  the  vulgar  fraction  cannot  be  exactly  ex- 
pressed by  a  decimal  (Art.  158). 

Let  it  be  required  to  reduce  ^  to  its  equivalent  decimal. 

We  find  the  equivalent  decimal  to  be 
.4166  -|-  (fee,  giving  6's,  as  far  as  we 
choose  to  continue  the  division. 

The  further  the  division  is  continued 
the  nearer  the  decimal  will  approach  to  the  true  value  of  the 
vulgar  fraction  ;  and  we  express  this  approach  to  equality  of 
value  by  saying,  that  if  the  division  be  continued  without  limits 
that  is,  to  infinity,  the  value  of  the  decimal  will  then  be  equal 
to  that  of  the  vulgar  fraction.     Thus,  we  also  say, 

.999  +,  continued  to  infinity  =  1, 
because  every  annexation  of  a  9  brings  the  value  nearer  to  1. 

164.  Let  us  now  examine  the  circumstances  under  which, 
in  the  reduction  of  a  vulgar  to  a  decimal  fraction,  the  division 
will  not  terminate. 

If  the  vulgar  fraction  be  first  reduced  to  its  lowest  terms, 
(which  we  suppose  to  be  done  in  all  the  cases  which  follow,) 
there  will  be  no  factor  common  to  its  numerator  and  denomi- 
nator. Now,  by  the  addition  of  O's  to  the  numerator  we  may 
increase  its  value  ten  times  for  every  0  annexed ;  that  is,  we 
introduce  into  the  numerator  the  two  factors  2  and  5  for  every 

Quest. — 163.  Can  a  vulgar  fraction  always  be  exactly  expressed  by 
a  decimal?  Can  five-twelfths?  If  we  continue  the  division,  does  the 
quotient  approach  to  the  true  value  ?  By  what  language  do  we  express 
this  fact?  164.  In  annexing  a  0  to  the  numerator,  what  factors  do  we  in- 
troduce into  it? 


176 


CIRCULATING    OR    REPEATING    DECIMALS. 


additional  0.  But  the  numerator  can  never  be  exactly  divi- 
ded by  the  denominator,  if  the  denominator  contains  any 
prime  factor  not  found  in  the  numerator  (Art.  107) :  hence 
it  can  never  be  so  divided,  if  the  denominator  contains  any 
prime  factor  other  than  2  or  5.  Hence,  to  determine  whether 
a  vulgar  fraction  in  its  lowest  terms  can  be  expressed  by  an 
exact  decimal, 

Decompose  the  denominator  into  its  prime  factors,  and  if 
there  are  any  factors  other  than  2  or  5,  the  exact  division  can- 
not be  made. 

EXAMPLES. 

1 .  Can  -^  be  exactly  expressed  by  decimals  ? 

OPERATION. 

25  =  5  X  5 ;  hence,  the  fraction 
can  be  exactly  expressed  by  a  deci- 
mal. 


25)  70  (.28 
50 


200 
200 


36)  50  (.1388 -f 
36 


2.  Can  3^  be  exactly  expressed  by  decimals? 

OPERATION. 

36  =  18x2  =  9x2x2  = 
3x3x2x2;  in  which  we  see  that 
the  denominator  contains  other  fac- 
tors than  2  and  5,  and  hence  the 
fraction  cannot  be  exactly  expressed 
by  decimals. 


140 
108 
~320 

288 

320 

288 


3.  Can  j^  be  exactly  expressed  by  decimals  ? 

4.  Can  -ji^  be  exactly  expressed  by  decimals  ? 

5.  Can  3^5  be  exactly  expressed  by  decimals  ? 

6.  Can  illo  ^^  exactly  expressed  by  decimals  1 


Quest. — Under  what  circumstances  will  the  numerator  be  exactly  divisi- 
ble by  the  denominator?  When  not  so?  How  do  you  determine  whether 
a  vulgar  fraction  can  be  exactly  divisible  by  a  decimal  ? 


CIRCULATING    OR    REPEATING    DECIMALS.  177 

Note. — 165.  When  there  are  no  prime  factors  in  the  denomi- 
nator other  than  2  or  5,  the  division  will  always  be  exact,  and 
the  number  of  decimal  places  in  the  quotient  will  be  equal  to  the 
greatest  number  of  factors  among  the  2's  or  5's. 

7.  What  is  the  decimal  corresponding  to  the  fraction  y|^  ? 

8.  What  is  the  decimal  corresponding  to  -g^j  ? 

9.  What  is  the  decimal  corresponding  to  ^^  ? 

166.  The  decimals  which  arise  from  vulgar  fractions, 
where  the  division  does  not  terminate,  are  called  circulating 
decimals^  because  of  the  continual  repetition  of  the  same 
figures.  The  set  of  figures  which  repeats,  is  called  a  rcpe- 
tend. 

167.  A  Single  Repetend  is  one  in  which  only  a  single 
figure  repeats,  as  f  =  .2222  + ,  or  |  r=  .3333  +  •  Such  repe- 
tends  are  expressed  by  simply  putting  a  mark  over  the  first 
figure  ;  thus,  .2222+  is  denoted  by  ^2  +,   and  .3333  +    by 

168.  A  Compound  Repetend  has  the  same  figures  circula- 
ting alternately:  thus  i|  =  .5757+  and  f|Jf  =.57235723  + 
are  compound  repetends,  and  are  distinguished  by  marking 
the  first  and  last  figures  of  the  circulating  period.  Thus 
.5757+  is  written  ^57^+,  and  .57235723+  is  written 
?5723''+. 

169.  A  Pure  Repetend  is  an  expression  in  which  there 
are  no  figures  except  the  repeating  figures  which  make  up 
the  repetend;  as  .3+,  .5+,  .473^+,  &c. 

170.  A  Mixed  Repetend  is  one  which  has  significant 
figures  or  ciphers  between  the  repetend  and  the  decimal 

Quest. — 165.  If  there  are  no  prime  factors  in  the  denominator  other 
than  2  and  5,  will  the  division  be  exact  ?  How  many  decimal  places  will 
there  be  in  the  quotient?  166.  What  are  the  decimals  called  when  the 
division  does  not  terminate  ?  What  is  the  set  of  figures  which  repeats 
called?  167.  What  is  a  single  repetend?  How  is  it  expressed?  168.  What 
u«  a  compound  repetend?  How  is  it  expressed?  169.  What  is  a  purt* 
repetend?     170.  What  is  a  mixed  repetend? 

8* 


178  REDUCTION    OF    CIRCULATING    DECIMALS. 

point,  or  which  has  whole  numbers  at  the  left  hand  of  the 
decimal  point :  such  fissures  are  called  finite  figures.  Thus, 
.0\  +  ,  .0  733^+,  .4  73'+,  .3"573V,  6^-5,  and  4^375'+  are 
all  mixed  repetends,  .0,  .4,  .3,  and  6  are  the  finite  figures. 

171.  Similar  Repetends  are  such  as  begin  at  an  equal 
distance  from  ^he  decimal  points  ;  as  .3  54  -}-,  2.7  534  +. 

17^.  Dissimilar  Repetends  are  such  as  begin  at  different 
places  from  the  decimal  point;  as   .253  +,  .47  52  +. 

173.  Conterminous  Repetends  are  such  as  end  at  the 
same  distance  from  the  decimal  points  ;  as  .1^5  -}-,  .354  +, 
&c. 

174.  Similar  and  Conterminous  Repetends  are  such  as 
begin  and  end  at  the  same  distance  from  the  decimal  point : 
thus,  53.2^^753''  +  ,  4.6^325^+,  and  .4^632^+,  are  similar 
and  conterminous  repetends. 


reduction  of  circulating  decimals. 


175.  To  reduce  a  pure  repetend  to  its  equivalent  vulgar 
fraction. 

Since  J  =  .1  +,  and  f  =  .3  +,  and  |-|  =  .54^-f  ;  and 
since  all  repetends  may  be  placed  under  similar  forms  ;  there- 
fore, to  find  the  finite  value  of  a  pure  repetend, 

Make  the  given  repetend  the  numerator,  and  write  a  denomi- 
nator containing  as  many  9V  as  there  are  places  in  the  repetend, 
and  this  fraction  reduced  to  its  lowest  terms  will  he  the  equiva- 
lent fraction  sought. 

Quest. — What  are  such  figures  called?  171.  What  are  similar  repe- 
tends? 172.  What  are  dissimilar  repetends  ?  173.  What  are  conterminous 
repetends?  174.  What  are  similar  and  conterminous  repetends?  175 
How  do  you  reduce  a  pure  repetend  to  its  equivalent  vulgar  fraction  ? 


REDUCTION    OF    CIRCULATING    DECIMALS.  179 

EXAMPLES. 

1.  What  is  the  equivalent  vulgar  fraction  of  the  repetend 
0^3  +  ? 

Now,  i  =  i  =  0.33333  +.  =  0?3  +. 

2.  What  is  the  equivalent  vulgar  fraction  of  the  repetend 
."l62'+  ? 

We  have,  iff  =  j\\  Ans, 

3.  What  are  the  simplest  equivalent  vulgar  fractions  of  the 
repetends  ^6  +,  J62'+,  0>69230'+,  ^945"+,  and  ?09'+ ? 

4.  What  are  the  least  equivalent  vulgar  fractions  of  the 
repetends  ?594405'+,  M'+,  and  "l42857'+ ? 

CASE    IT. 

176.  To  reduce  a  mixed  repetend  to  its  equivalent  vulgar 
fraction. 

A  mixed  repetend  is  composed  of  the  finite  figures  which 
precede,  and  of  the  repetend  itself;  and  hence  its  value  must 
be  equal  to  such  finite  figures  plus  the  repetend.  Hence,  to 
find  such  value. 

To  the  finite  figures  add  the  repetend  divided  hy  as  many  9^s 
as  it  contains  places  of  figures,  with  as  many  OV  annexed  to 
them  as  there  are  places  of  decimal  figures  which  precede  the 
repetend ;  the  sum  reduced  to  its  simplest  form  will  be  the  equiv- 
alent fraction  sought. 

EXAMPLES. 

1.  Required   the  least  equivalent  vulgar  fraction  of  the 
mixed  repetend  2.4  18  +. 
Now, 

2.4^1 8^+  :::,  2  +  /^  +  ^^  +  =  2  +  3^  +  9'^  =  2|f  Ans. 

Quest. — 176.  How  do  you  reduce  a  mixed  repetend  to  its  equivalen* 
vulgar  fraction  ? 


180  REDUCTION    OF    CIRCULATING    DECIMALS. 

2.  Required  the  least  equivalent  vulgar  fraction  of  the 
mixed  repetend   .5  925  +. 

We  have,    .5^925'+  =  -3^  +  oVo^  =  if  -^^^• 

3.  What  is  the  least  equivalent  vulgar  fraction  of  the  repe- 
tend .008  497133^+? 

We  have,     .008  497133'+  ==  y^^  +  9  qWfq'^  =  ofls  • 

4.  Required  the  least  equivalent  vulgar  fractions  of  the 
mixed  repetends  .13"8 +,  7.5\3'+,  .04"354'+,  37.5^4  +  , 
.6  75'+,  and   .7"54347'+. 

5.  Required  the  least  equivalent  vulgar  fractions  of  the 
mixed  repetends  0.7^5 -|-,  0.4^38''+,  .09^3+,  4.7''543'+, 
,009  V  +  ,  and    .4^5  -f . 

177.  There  are  some  properties  of  the  repetends  which  it 
is  important  to  remark. 

1.  Any  finite  decimal  may  be  considered  as  a  circulatmg 
decimal  by  making  ciphers  to  recur :  thus, 

.35  zz:  .35  0  +  =  .35^00^  =  .35^000^+  =  .35^000^+,  &c. 

2.  If  any  circulating  decimal  have  a,  repetend  of  any  num- 
ber of  figures,  it  may  be  reduced  to  one  having  twice  or  thrice 
that  number  of  figures,  or  any  multiple  of  that  number. 

Thus,  a  repetend  2.3  57^+,  having  two  figures,  may  be  re- 
duced to  one  having  4,  6,  8,  or  10  places  of  figures.  For, 
2.3W  ==2.3"5757'+  =  2.3^5757"+  =  2.3  57575757'+ 
&c. ;  so,  the  repetend  4.16316  +  may  be  written 
4.16"316'+  =z  4.16  316316'+  =4.16  31631 6316'+  &c.  &c. , 
and  the  same  may  be  shown  of  any  other.  Hence,  two  or 
more  repetends,  having  a  different  number  of  places  in  each, 
may  be  reduced  to  repetends  having  the  same  number  of 
places,  in  the  following  manner : 

Quest. — 177.  What  is  the  first  property  of  the  circulating  decimals? 
How  do  you  reduce  several  repetends  having  different  places  in  each,  to 
repetends  having  the  same  number  of  places  ? 


REDUCTION    OF    CIRCULATING    DECIMALS.  181 

Find  the  least  common  multiple  of  the  number  of  places  in 
each  repetend,  and  reduce  each  repetend  to  such  number  of  places. 

1.  Reduce  .13'8  +  ,  7.5  43^+,  .04  354^  to  repetends 
having  the  same  number  of  places. 

Since  the  number  of  places  are  now  1,  2,  and  3,  the  com- 
mon multiple  will  be  6,  and  hence  each  new  repetend  will 
contain  6  places.     Hence, 

.IS'^S  +  r=  .13^888888^4-  ;    7.5  43^+  =  7.5  434343^+  ; 
0.4'354'+  =  0.4  354354^+. 

2.  Reduce  2.4"l8'+,  .5V25'4-,  .008497133''+  to  repe- 
tends having  the  same  number  of  places. 

3.  Any  circulating  decimal  may  be  transformed  into  an- 
other having  finite  decimals  and  a  repetend  of  the  same 
number  of  figures  as  the  first.     Thus, 

.57'+  =  .5  75'+  =  .57V+=  .575W+=  .575757';  and 
3.4>85'+  =  3.47^857'+  =  3.478"578'+  =  3.4785  785'+  ; 
and  hence,  any  two  repetends  may  be  made  similar.  These 
properties  may  be  proved  by  changing  the  repetends  into  their 
equivalent  vulgar  fractions. 

4.  Having  made  two  or  more  repetends  similar  by  the  last 
article,  they  may  be  rendered  conterminous  by  the  previous 
one :  thus,  two  or  more  repetends  may  always  he  made  similar 
and  conterminous. 

1.  Reduce  the  circulating  decimals  165.164  +,  .04  +, 
.03  7  +  to  such  as  are  similar  and  conterminous. 

"^  ^    ^ 

2.  Reduce  the  circulating  decimals  .5  3  +,  .4  75  +,  and 

1.757  +,  to  such  as  are  similar  and  conterminous. 

5.  If  two  or  more    circulating  decimals,  having  several 

Quest. — When  a  repetend  has  more  than  one  figure,  may  it  be  trans- 
formed into  a  circulating  decimal  having  finite  decimals?  How  many- 
places  must  there  be  in  the  repetend  ?  What  are  similar  and  conterminous 
repetends  ?    May  all  circulating  decimals  be  made  similar  and  conterminous  ? 


182  REDUCTION    OF    CIRCULATING    DECIMALS. 

repetends  of  equal  places,  be  added  together,  their  sum  will 
have  a  repeterid  of  the  same  number  of  places  ;  for  every  two 
sets  of  repetends  will  give  the  same  sum.  ' 

6.  If  any  circulating  decimal  be  multiplied  by  any  number, 
the  product  will  be  a  circulating  decimal  having  the  same 
number  of  places  in  the  repetend  ;  for,  each  repetend  will  be 
taken  the  same  number  of  times,  and  consequently  must  pro- 
duce the  same  product. 


178.  To  find  the  number  of  places  in  the  repetend  cor- 
responding to  any  vulgar  fraction  which  cannot  be  expressed 
by  a  finite  decimal. 

Let  the  fraction  be  first  reduced  to  its  lowest  terms,  after 
which  find  all  the  prime  factors  2  and  5  of  the  denominator. 
Then  separate  the  fraction  into  two  factors,  viz., 

1st.  The  numerator  divided  by  the  product  of  all  the  prime 
factors  2  and  5  ;  and 

2d.  Unity  divided  by  the  remaining  factor  of  the  denomi- 
nator. 

As  an  example,  let  us  decompose  the  fraction  -g^  into  the 
two  factors  named  above.     They  are, 

3    _  3  1 

280"~2X2X2X5^Y* 
If,  now,  we  add  a  0  to  the  1  and  proceed  to  make  the  division, 
every  remainder  will  be  less  than  the  divisor,  and  hence  we 
cannot  make  more  divisions  than  there  are  units  in  the  divi- 
sor less  1,  without  reducing  the  remainder  to  unity,  when  the 
first  quotient  figures  will  repeat.  And  observe  carefully  when 
any  remainder  becomes  the  same  as  a  remainder  previously 
usedjfor  at  this  point  the  repeating  jigures  begin. 

Quest. — ^Whatisthe  fifth  property  named ?  What  is  the  sixth?  178. 
What  is  the  first  operation  in  finding  the  form  of  the  decimal  corresponding 
to  a  given  vulgar  fraction  ?  Into  how  many  factors  is  it  then  divided  ? 
What  are  these  factors  ?  How  many  divisions  may  be  performed  in  the 
second  factor? 


,  REDUCTION    OF    CIRCULATING    DECIMALS.  183 

If,  now,  we  suppose  the  remainder  1  to  be  subtracted  from 
the  dividend  so  used,  there  would  remain  as  many  9's  as 
there  were  divisions.     Hence, 

If,  after  having  taken  out  the  2^s  and  5^ s  from  the  denomina- 
tor, we  divide  a  succession  of  9'^  by  the  result  until  there  is  no 
remainder,  the  number  of  9'^  so  used  will  be  equal  to  the  number 
of  places  of  the  repetend,  which  can  never  exceed  the  number  of 
units  in  the  denominator  less  one. 

Having  found  the  number  of  finite  decimals  which  precede 
the  repetend,  and  the  number  of  places  in  the  repetend,  as 
above. 

Divide  the  numerator  of  the  vulgar  fraction,  reduced  to  its 
lowest  terms,  by  the  denominator,  and  point  off  in  the  quotient 
the  finite  decimals,  if  any,  and  the  repetend, 

EXAMPLES.  * 

1.  Required  to  find  whether  the  decimal  equivalent  to 
sJ^rll  ^®  finite  or  circulating ;  the  number  of  places  in  the 
repetend  and  the  place  at  which  the  repetend  begins ;  and, 
also,  the  equivalent  circulating  decimal. 


We  first  reduce  the  fraction 
to  its  lowest  terms,  giving  g-lfs  • 
We  then  search  for  the  factors 
2  and  5  in  the  denominator,  and 
find  that  2  is  a  factor  3  times : 
hence  we  know  that  there  are 
three  finite  decimals  preceding 
the  repetend.  We  next  divide 
99999,  &c.,  by  the  factor  1221 
of  the   denominator,   and  find 


OPERATION. 

249      83 

29304    —   9  76^ 

2)9768 
2)4884 
2)2442 


1221 

1221)999999(819 

^f^  =.  .008497133^ 

that  we  use   six  nines  before   the  remainder  becomes   0 : 
hence,  we  know  that  there  are  six  places  of  figures  in  the 

Quest. — What  will  determine  the  highest  limit  of  the  number  of  figures 
in  the  repetend?  What  will  dete^rmine  the  number  of  finite  decimals? 
How  then  will  you  find  the  equivalent  decimal  ? 


184  ADDITION    OF    CIRCULATING    DECIMALS. 

repetend.     We  then  divide  83  by  9768,  and  point  off  the 
proper  places  in  the  quotient. 

2.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if 
any,  of  the  fraction  yyTo' 

3.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if 
any,  of  the  fraction  ytVo  • 

4.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if 

1  2  80       _7_2_ 

L23'     133'     135* 


any,  of  the  fractions  ^^    -^^    -^-^- 


ADDITION    OF    CIRCULATING    DECIMALS. 

179.  To  add  circulating  decimals. 

Make  the  repetends  similar  and  conterminous,  and  then  write 
the  places  of  like  value  under  each  other,  and  so  many  figures  of 
the  second  repetends  to  the  right  as  shall  indicate  with  certainty 
how  moffiy  are  to  he  carried  from  one  repetend  to  the  other ;  after 
which  add  them  as  in  whole  numbers.  If  all  the  figures  of  a 
repetend  he  9'^,  omit  them  and  add  one  to  the  figure  next  to  the 
\eft. 

EXAMPLES, 

1.  Add  ,Ylb  +,  4."l63'+,   l"7143^,  and  2.54^  together. 
Dissimilar.  Similar.  Similar  and  conterminous. 

.12"i5  +  =    .12  5  +        =   .12^555555555555^+    .   .   5555 

4.''l63'+    =4. 16^316''+    i=4. 16  316316316316'+   .   .  3163 

1.7143'+=:1.71437l'+  =  1.7143714371437l'+   .   .  4371 

2.M''+      =2. 54^54'+      :=  2. 54^^545454545454''+    .   .  54=54 

The  true  sum    r^ 8. 54^8544701 31 697'+,  1  to  carry. 

2.  Add  67.3  45'+,  9?65l'+,  ^25'  +  ,  17.4>+,  ^5 +"  to- 
gether. 

3.  Add  ^475'+,  3.75^43'  +  ,  64>5'+,  .^57'+,  .r788'  + 
together. 

Qttest  — 1 79.  How  do  yon  add  cironlatinjy  decimals  ? 


SUBTRACTION    OF    CIRCULATING    DECIMALS.  185 

4.  Add  .5+,  4.3>+,  49.4  57'+,  .4"954'+,  .7345'+ 
together. 

5.  Add  "175'+,  42.57'+,  .3>53'+,  .5"945'+.  3.7^54'+ 
together. 

6.  Add  165,  ."164' +,  147?04'+,  4.95'+,  94.3  7+, 
4.>123456'+  together. 

SUBTRACTION    OF    CIRCULATING    DECIMALS. 

180.  To  subtract  one  finite  decimal  from  another. 

Make  the  repetends  similar  and  conterminous,  and  subtract 
as  in  finite  decimals,  observing  that  when  the  repetend  of  the 
lower  line  is  the  largest  its  first  right  hand  figure  must  be  in- 
creased by  unity, 

EXAMPLES. 

1.  From  11.4>5'+  take  3.45  735'+. 
Dissimilar.  Similar.         Similar  and  conterminous. 

11. 4^75'+       =  11.4757'+     =  11.47575757'+    ...  575 

3.45735'+  =    3.45  735'+  z^     3.45>35735'+    ...  735 

•  The  true  difference  =    8.01  84002l'+,  1  to  carry. 

2.  From  47.5^3  +  take  1/757^+.  Ans.  

3.  From  17.^573'+  take  14.5>  +.  Ans,  

4.  From  17.4  3  +  take  12.34  3  +.  Ans.  

5.  From  1.12>54'+  take  .4  7384'+.  Ans,  

6.  From  4.75  take   .37'5  +.  Ans.  

7.  From  4.794  take  .1744'+.  Ans.  

8.  From  1.45>+  take  .3654.  Ans,  

9.  From  1.4^^937'+  take  .1475.  Ans,  

Quest. — 180.  How  do  you  subtract  circulating  decimals? 


186       MULTIPLICATION    OF    CIRCULATING    DECIMALS 
MULTIPLICATION    OF    CIRCULATING    DECIMALS. 

181.  To  multiply  one  circulating  decimal  by  another. 

Change  the  circulating  deximals  into  their  equivalent  vulgar 
fractions,  and  then  multiply  them  together ;  after  which  reduc6 
the  product  to  its  equivalent  circulating  decimal,  as  in  Art.  178. 

EXAMPLES. 

1.  Multiply  4.25^3+  by  .257. 

OPERATION. 

4.25  3  4-  —  4.-2-^  -4-  -^-  —  4  4-  ^^^  -A-  -^  --  223  — 

T,^^  xf     I     —  ^'100     >     900    —  ^     i     900      '     900    — — 


900 


- 1914   957 

-   450     —  225" 


Also,  .257  =  y2_5^  .  hence, 

957    V      25  7     _    2jL5^19l   _   1    OQ'^lO^fi   4-  • 
^T5     ^    TOOO   —    225000    —    i-.^VOlKJU    -f-  , 

and  since  225000  =  5x5x5x5x5x2x2x2x9, 
there  will  be  five  places  of  finite  decimals,  and  one  figure  in 
the  repetend  (Art.  178). 

Note.  Much  labor  will  be  saved  in  this  and  the  next  rule  by 
keeping  every  fraction  in  its  lowest  terms,  and  when  two  fractions 
are  to  be  multiplied  together,  cancelling  all  the  factors  common  to 
both  terms  before  making  the  multiplication. 

Q.  Multiply  .375\  +  by  14.75.  Ans,  — '- 

3.  Multiply  .4"253'+   by  2.57.  Ans.  

4.  Multiply  .437  by  3.7  5  4-.  Ans.  

5.  Multiply  4.573  by.3>5^+.  Ans.  

6.  Multiply  3-45  6 -f  by  .42  5+.  Ans.  

7.  Miiltiply  L456'+  by  4.2^3  +.  Ans,  

8.  Multiply  45. r3  +  by  ?245'+.  Ans. 

9.  Multiply  .4705^3  +  by  1.7^35^+.  Ans.  

10.  Multiply  3.457^3  +  by  54.753''+.  Ans.  

Quest. — 181.  How  do  you  multiply  circulating  decimals?  What  is  to 
be  observed  in  regard  to  keeping  fractions  in  their  lowest  terms  ? 


DIVISION    OF    CIRCULATING    DECIMALS.  187 

DIVISION    OF    CIRCULATING    DECIMALS. 

182.  To  divide  one  circulating  decimal  by  another. 

Change  the  decimals  into  their  equivalent  vulgar  fractions^ 
and  find  the  quotient  of  these  fractions.  Then  change  the  quo 
tient  into  its  equivalent  decimal,  as  in  Art.  178. 

EXAMPLES. 

1.  Divide  56j5  +  by  137. 

OPERATION. 
56^6  -f  =:  56  +  f  =.:  ^  =  11.0. 
Then,    i|o  ^  137  ^  ijo  X  yly  =  ^1^  =  ."41362530'+. 

2.  Divide  24.3^^1 8^  by  1.792.  Ans,  

3.  Divide  8.5968  by  .2^5^+.  Arts,  

4.  Divide  2.295  by  ?297V.  Ans. 

5.  Divide  47.345  by  1?76'+.  Ans,  

6.  Divide  13.5"l69533'+  by  4.297^  +  .  Ans,  

7.  Divide  ?45'+  by  .118881'  +  .  Ans,  

8.  Divide  ?475V  by  .3>53V.  Ans.  . 

9.  Divide  3>53'  by  ^^24"+.  Ans,  

Quest. — 182.  How  do  you  divide  circulating  decimals? 


188  RATIO    AND    PROPORTION    OF    NUMBERS 


OF  THE  RATIO  AND  PROPORTION  OF  NUMBERS 

183.  Two  numbers  having  the  same  unit  may  be  comparecj 
together  in  two  ways. 

1st.  By  considering  how  much  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference  ;  and 

2d.  By  considering  how  many  times  one  is  greater  or  less 
than  the  other,  which  is  shown  by  their  quotient. 

Thus,  in  comparing  the  numbers  3  and  12  together  with 
respect  to  their  difference,  we  find  that  12  exceeds  3  by  9 ; 
and  in  comparing  them  together  with  respect  to  their  quo- 
tient, we  find  that  12  contains  3  four  times,  or  that  12  is  four 
times  as  great  as  3. 

The  quotient  which  arises  from  dividing  the  second  num- 
ber by  the  first,  is  called  the  ratio  of  the  numbers,  and  shows 
how  many  times  the  second  number  is  greater  than  the  first, 
or  how  many  times  it  is  less. 

Thus,  the  ratio  of  3  to  9  is  3,  since  9-^-3  =  3.  The  ratio 
of  2  to  4  is  2,  since  4-^-2=2. 

We  may  also  compare  a  larger  number  with  a  less.  For 
example,  the  ratio  of  4  to  2  is  ^  ;  for,  2  ~  4  =  i.  The  ratio 
of  9  to  3  is  i,  since  3-^9=^. 

EXAMPLES. 

1.  What  is  the  ratio  of  9  to  18  ?  Ans.  

2.  What  is  the  ratio  of  6  to  24  ?  Ans.  


Quest. — 183.  In  how  many  ways  may  two  numbers  having  the  same 
miit  be  compared?  How  do  you  determine  how  much  one  number  is 
greater  than  another  ?  How  do  you  determine  how  many  times  it  is  greater 
or  less?  How  much  does  12  exceed  3?  How  many  times  is  12  greater 
than  3  ?  What  is  the  quotient  called  which  arises  from  dividing  the  second 
number  by  the  first  ?  What  does  it  show  ?  When  the  second  number  is 
the  least,  what  does  it  show  ? 


RATIO    AND    PROPORTION    OF    NUMBERS.  189 

3.  What  is  the  ratio  of  12  to  48  ?  Ans.  — — 

4.  What  is  the  ratio  of  11  to  13  ?  Ans.  

5.  What  part  of  20  is  4  ?    Or  what  is  the  ratio  of  20  to  4  ? 

6.  What  part  of  100  is  30  ?  Or  what  is  the  ratio  of  100 
to  30? 

7.  What  part  of  6  is  3  ?  Ans.  

8.  What  part  of  9  is  3  ?  Ans.  

9.  What  part  of  12  is  4  ?  Ans.  

10.  What  part  of  50  is  5  ?  Ans.  

11.  What  part  of  75  is  3  ?  .  Ans.  

Note. — In  determining  what  part  one  number  is  of  another,  it  is 
plain  that  the  number  to  be  measured,  must  be  written  in  the  nu- 
merator; while  the  standard,  or  number  with  which  it  is  compared, 
and  of  which  it  forms  a  part,  is  written  in  the  denominator.  This 
fraction,  reduced  to  its  lowest  terms,  will  express  the  part. 

184.  If  one  yard  of  cloth  cost  $2,  how  many  dollars  will 
6  yards  of  cloth  cost  at  the  same  rate  1 

It  is  plain  that  6  yards  of  cloth  will  cost  6  times  as  much 
as  one  yard ;  that  is,  the  cost  will  contain  $2  as  many  times 
as  6  contains  1.     Hence  the  cost  will  be  $12. 

In  this  example  there  are  four  numbers  considered,  viz., 
1  yard  of  cloth,  6  yards  of  cloth,  f  2,  and  $12  :  these  num- 
bers are  called  terms, 

1  yard  of  cloth  is  the       1st  term, 
6  yards  of  cloth  is  the     2d  term, 
$2  is  the        -       -       -      3d  term, 
$12  is  the        -       -       -      4th  term. 
Now  the  ratio  of  the  first  term  to  the  second  is  the  same  as 
the  ratio  of  the  third  to  the  fourth. 

This  relation  between  four  numbers  is  called  a  proportion ; 
and  generally 

Four  numbers  are  said  to  be  in  proportion  when  the  ratio  of 

Quest. — How  do  you  determine  what  part  one  number  is  of  another? 
184.  If  one  yard  of  cloth  cost  $2,  what  will  6  yards  cost?  How  many 
numbers  are  here  considered  ?  What  are  they  called  ?  What  is  the  ratio 
of  the  first  to  the  second  equal  to?  What  is  this  relation  between  numbers 
called  '     When  are  four  numbers  said  to  be  in  proportion  ? 


190 


RATIO    AND    PROPORTION    OF    NUMBERS. 


the  first  to  the  second  is  the  same  as  that  of  the  third  to  the 
fourth.     Hence, 

A  Proportion  is  an  equality  of  ratios   between  numbers 
compared  together  two  and  two. 

185.  We  express  that  four  numbers  are  in  proportion  thus : 

1     :     6     :    :     2     :     12. 
That  is,  we  write  the  numbers  in  the  same  line  and  place 
two  dots  between  the  1st  and  2d  terms,  four  between  the  2d 
and  3d,  and  two  between  the  3d  and  4th  terms.     We  read  the 
proportion  thus, 

as  1  is  to  6,  so  is  2  to  12. 
The  \st  and  2d  terms  of  a  proportion  always  express  quanti- 
ties of  the  same  kind,  and  so  likewise  do  the  Zd  and  4th  terms. 
As  in  the  example, 

yd.       yd.  $  $ 

1     :     6     :  :     2     :     12. 
This  is  implied  by  the  definition  of  a  ratio  ;  for,  it  is  only- 
quantities  of  the  same  kind  which  can  be  divided  the  one  by 
the  other.    The  ratio  of  the  first  term  to  the  second,  or  of  the 
third  to  the  fourth,  is  called  the  ratio  of  the  proportion. 
1 .  What  are  the  ratios  of  the  proportions 


3  : 

9  : 

:  12 

36? 

Ans. 

2  : 

10  : 

12  . 

60? 

Ans. 

4  : 

2  :  , 

8 

4? 

Ans. 

9  : 

1  : 

90 

10? 

Ans. 

16  : 

15  :  . 

48  . 

45? 

Ans. 

186.  When  two  numbers  are  compared  together,  the  first 
is  called  the  antecedent,  and  the  second  the  consequent ;  and 
when  four  numbers  are  compared,  the  first  antecedent  and 
consequent  are  called  the  first  couplet,  and  the  second  ante- 
cedent and  consequent  the  second  couplet.     Thus,  in  the  last 

Quest. — How  do  you  define  proportion?  185.  How  do  you  indicate 
that  four  narabers  are  in  proportion?  How  is  the  proportion  read?  What 
do  you  remark  of  the  first  and  second  terms  ?  Also  of  the  third  and  fourth  I 
186.  When  two  numbers  are  compared  together,  what  is  the  first  called? 
What  the  second?  When  four  numbers  are  compared,  what  are  the  two 
first  called  ?     What  the  two  second  ? 


RATIO    AND    PROPORTION    OF    NUMBERS. 


191 


proportion,  16  and  48  are  the  antecedents,  and  15  and  45  the 
consequents  ;  also,  16  and  15  make  the  first  couplet,  and  48 
and  45  the  second. 

187.  We  have  said  that  proportion  is  an  equality  of  ratios 
(Art.  184).  Besides  the  method  above,  we  may  express  that 
equality  thus : 

4_  6^ 

2~"  3' 
and  we  may  then  write  the  proportion  thus : 
2     :     4     :  :     3     :     6. 
Put  the  following  equal  ratios  into  proportion. 


8        16 

21        105 

*     9  ~  18' 

•    16  -   80  * 

17       19 

42   .   252 

*  '    51  ~"57' 

•    35-210' 

9        27 

29       232 

^'    16-48- 

'    37-296' 

19       76 

45       405 

*    13-52* 

•    23-207' 

188.  If  4:1b.  of  tea  cost  $8,  what  will  I2lb.  cost  at  the  same 

rate? 

' 

OPERATION. 

lb.     lb,        $ 

$ 

As  4  :   12  :  :  8  : 

Ans. 

12 
4)96 

i^x8  =  3x8=24 

4 

$24  the  cost  of  12lb. 

of  tea.                            Ans.  $24 

It  is  evident  that  the  4th  term,  or  cost  of  I2lb.  of  tea,  must 
be  as  many  times  greater  than  $8,  as  I2lb.  is  greater  than 
4:1b.  But  the  ratio  of  4lb.  to  I2lb.  is  3  ;  hence,  3  is  the  num- 
ber of  times  which  the  cost  exceeds  $8 :  that  is,  thS  cost  is 


Quest. — 187.  What  has  proportion  been  called  ?   By  what  second  method 
»Tay  this  equality  be  expressed  ?     188.  Explain  this  example  mentally. 


192      RATIO  AND  PROPORTION  OF  NUMBERS. 

equal  to  $8  x  3  =  $24.     But  instead  of  writing  the  numbers 

12 

Y  X  8  =  24, 

we  may  write  them 

(12   X  8  )  -^4  =  24: 
and  as  the  same  may  be  shown  for  every  proportion,  we  con- 
clude, 

That  the  4th  term  of  every  'proportion  may  he  found  by  mul- 
tiplying the  2d  and  ^d  terms  together,  and  dividing  their  product 
by  the  \st  term. 

EXAMPLES. 

1.  The  first  three  terms  of  a  proportion  are  1,  2,  and  3 : 
what  is  the  fourth  ?  Ans.  

2.  The  first  three  terms  are  6,  2,  and  1  :  what  is  the  4th? 

Ans.  

3.  The  first  three  terms  are  10,  3,  and  1  :  what  is  the  4th  ^ 

Ans.  

189.  The  1st  and  4th  terms  of  a  proportion  are  called  the 
two  extremes,  and  the  2d  and  3d  terms  are  called  the  two 
means. 

Now,  since  the  4th  term  is  obtained  by  dividing  the  product 
of  the  2d  and  3d  terms  by  the  1st  term,  and  since  the  product 
of  the  divisor  by  the  quotient  is  equal  to  the  dividend,  it 
follows. 

That  in  every  proportion  the  product  of  the  two  extremes  is 
equal  to  the  product  of  the  two  means. 

Thus,  in  the  example.  Art.  184  we  have 

1  X  12  =    2  X     6; 


also. 


1  : 

6  : 

:  2 

12; 

and 

4  : 

12  : 

:  8 

24; 

and 

6  : 

9  : 

:  10 

15; 

and 

7  : 

15  : 

:  14 

30; 

and 

4  X  24  =  12  X     8 

6  X  15  =    9  X  10 

7  X  30  =:  15  X  14. 


Quest. — How  may  the  fourth  term  of  eveiy  proportion  be  found?  189 
What  are- the  first  and  fourth  teiTns  of  a  proportion  called?  What  are  the 
second  and  third  terms  called  ?  In  every  proportion,  what  is  the  product 
of  the  extremes  equal  to? 


OF  CANCELLING.  193 


OF  CANCELLING. 

190.  When  one  number  is  to  be  divided  by  another,  the 
operation  may  often  be  shortened  by  striking  out  or  cancelling 
the  factors  common  to  both,  before  the  division  is  made. 

1.  For  example,  suppose  it  were  required  to  divide  360 
by  120. 

We  first  write  the 
dividend  above  a  ho- 
rizontal line,  and  the 


OPERATION. 

360       12x30      X%X3X0 


120       12  X  10         /rX  x/0     ""    ' 
divisor  beneath  it,  af-     ' 

ter  the  form  of  a  fraction.     We  next  separate  both  of  them 
into  factors,  and  then  cancel  the  factors  which  are  alike. 

2.  Divide  630  by  35. 

We   separate  the  divi-  operation. 

dend  and  divisor  into  like  630  _3  X/OX  6  X''y__ 

factors,   and  then    cancel  35  """         ^i  xX  ~" 

those  which  are  common  in  both. 


3.  Divide  1860  by  36. 
4    Divide  7920  by  720. 

5.  Divide  1890  by  210. 

6.  Divide  1260  by  504. 


7.  Divide  1768  by  221. 

8.  Divide  2856  by  238. 

9.  Divide  3420  by  285. 
10.  Divide  9072  by  1512. 


191.  If  two  or  more  numbers  are  to  be  multiplied  together 
and  their  product  divided  by  the  product  of  other  numbers, 
the  operation  may  be  abridged  by  striking  out  or  cancelling 
any  factor  which  is  common  to  the  dividend  and  divisor.  For 
example,  if  6  is  to  be  multiplied  by  8  and  the  product  divided 
by  4,  we  have 

^  X  S       48       ^^  6  X  8       ^       ^       _ 

-^=-  =  12;    or,    __  =  6X2  =  12: 


Quest. — 190.  How  may  the  division  of  two  numbers  be  often  abridged? 
Explain  the  example  mentally.  Also  the  second  example.  191.  When 
two  numbers  are  multiplied  together  and  their  product  divided  ^y  a  thuxL 
how  may  the  operation  be  abridged  ? 

9 


194  OF    CANCELLING. 

in  the  latter  case  we  cancelled  the  factor  4  in  the  numeratoi 
and  denominator,  and  multiplied  6  by  the  quotient  2. 

1.  Let  it  be  required  to  multiply  24  by  16  and  divide  the 
product  by  12.  ,      operation. 

Having  written  the  product  of  the  fig-      |      2 
ures,  which  form  the  dividend,  above  ^^  X  16 

the  line,  and  the  product  of  the  figures      ;  ~X^      ^ 

which  form  the  divisor  below  it,  then  i  1 

We  cancel  the  common  factors  in  the  numerator  and  denomi- 
nator^ and  write  the  quotients  over  and  under  the  numbers  in 
which  such  common  factors  are  found,  and  if  the  quotients  still 
have  a  common  factor,  they  v^ay  he  again  divided. 

2.  Reduce  the  compounci  fraction  |  of  |  of  ^^  of  ^-^  to  a 
simple  fraction. 

Beginning  with  the  first  nu- 
merator, we  find  it  is  once  a 
factor  of  itself  and  4  times  in 
16  ;  Q  is  twice  a  factor  in  12  ; 
3  three  times  a  factor  in  9  ; 
and  5,  once  a  factor  in  the  denominator  5. 


OPERATION. 
1111 

13        2         4 


3.  What  is  the  quotient  of3x8x9x7xl5  divided 
by  63  X  24  X  3  X  5  ? 

This  example  presents  a      j  operation. 

case    that   often  arises,   in  Z  y.^  y.  ^  Y.^  X  ^^  _ 

which  the  -product  of  two  |  i^^  x  ^^  X  Z  X  ^  ~~  * 
factors   may  be   cancelled.      I 

Thus,  3  X  8  is  24 :  then  cancel  the  3  and  8  in  the  numera- 
tor and  the  24  in  the  denominator.  Again,  9  times  7  are 
63  ;  therefore  cancel  the  9  and  7  in  the  numerator  and  thf^ 
63  in  the  denominator.  Also,  3  X  5  in  the  denominator  can- 
ce  s  the  15  remaining  in  the  numerator:  hence,  the  quotient 
is  unity. 

4.  What   is   the  quotient  of    126  X  16  X  3  divided    by 
7  X  12? 


OP    CANCELLING. 

OPERATION. 


=  rs. 


We  see  that  7  is  a  factor  of 
126,  giving  a  quotient  18,  which 
we  place  over  126,  crossing  at  the 
same  time  126  and  the  7  below. 
We  then  divide  18  and  12  by  6, 
crossing  them  both  and  vnriting 
-down  the  quotients  3  and  3.  We 
next  divide  16  and  2  by  2,  giving  the  quotients  8  and  1. 
Hence,  the  result  is  72. 


3 

^^         8 
AM  X  IX  X  3 

1 


EXAMPLES. 

1.  What  is  thequotientof  1x6x9x14x15x7x8 
divided  by  36  X  128  X  56  X  20  ? 

2.  What  is  the  value  of  18  X  36  X  72  X  144  divided  by 
6x6x8x9x12x8? 

3.  What  is  thequotientof  3  X  9  X  7  X  3  X  14  X  36  di- 
vided by  252  X  81  X  2  X  7  ? 

4.  What  is  the  quotient  of   19  X  17  X  16  X  8  X  9  X  6 
divided  by  32  X  4  X  27  X  2  ? 

5.  What  is  thequotient  of  4  X  12  X  16  X  30  X  16  X  48  X  48 
divided  by  9  X  10  X  14  X  24  X  44  X  40  ? 

192.  The  process  of  cancelling  may  be  applied  to  the 
terms  of  a  proportion. 

If  we  have  any  proportion,  as 

6     ;     15     :   :     28     :     70, 

We  may  always  cancel  like  factors  in  either  couplet.     Thus, 

2  5  14  35 

6     :     15     :  :     28     :     70, 

in  which  we  divide  the  terms  of  the  first  couplet  by  3,  and 

those  of  the  second  by  2,  and  write  the  quotients  above. 

EXAMPLES. 

1.  What  is  the  simplest  form  of      18  :     72  :  :  100  :  400  ? 

2.  What  is  the  simplest  form  of      14  :     49  :  :    42  :  147  \ 

3.  What  is  the  simplest  form  of    365  :  876  :  :  140  :  336  ? 

Quest. — 192.  How  else  may  the  process  of  cancelling  be  applied?  Whai 
may  be  cancelled  in  each  couplet  t 


196  RULE    OF   THREE. 


RULE  OF  THREE. 

193.  The  Rule  of  Three  takes  its  name  from  the  circum- 
stance that  three  numbers  are  always  given  to  find  a  fourth, 
which  shall  bear  the  same  proportion  to  one  of  the  given 
numbers  as  exists  between  the  other  two. 

The  following  is  the  manner  of  finding  the  fourth  term  : 

I.  Reduce  the  two  numbers  which  have  different  names  from 
the  answer  sought,  to  the  lowest  denomination  named  in  either 
of  them* 

II.  Set  the  number  which  is  of  the  same  kind  with  the  answer 
sought  in  the  third  place,  and  then  consider  from  the  nature  of 
the  question  whether  the  answer  will  be  greater  or  less  than  the 
third  term. 

III.  When  the  answer  is  greater  than  the  third  term,  write 
the  least  of  the  remaining  numbers  in  the  first  place,  but  when 
it  is  less  place  the  greater  there. 

IV.  Then  multiply  the  second  and  third  terms  together,  and 
divide  the  product  by  the  first  term :  the  quotient  will  be  the 
fourth  term  or  answer  sought,  and  will  be  of  the  same  denomi' 
nation  as  the  third  term. 

EXAMPLES. 

1.  If  48  yards  of  cloth  cost  $67,25  what  will  144  yards 
cost  at  the  same  rate  ? 

Quest. — 193.  From  what  does  the  Rule  of  Three  take  its  name  ?  What 
is  the  first  thing  to  be  done  in  stating  the  question  ?  Which  number  do  you 
make  the  third  term  ?  How  do  you  determine  which  to  put  in  the  first  ? 
After  stating  the  question,  how  do  you  find  the  fourth  term  ?  What  will 
be  its  denomination  ? 


RULE    OF    THREE. 


197 


yd, 

48 


OPERATION. 

yd,  $ 

144    :  :    67,25 
144 


In  this  example,  as  the 
answer  is  to  be  dollars, 
we  place  the  $67,25  in 
the  third  place.  Then,  as 
144  yards  of  cloth  will 
cost  more  than  48  yards, 
the  fourth  term  must  be 
greater  than  the  third,  and 
therefore,  we  write  the 
least  of  the  two  remain- 
ing numbers  in  the  first 
place.  The  product  of 
the  second  and  third  terms 
is  $9684,00 :  dividing  this 
by  the  first  term,  we  ob- 
tain $201,75  for  the  cost  of  144  yards  of  cloth. 


Ans 


26900 
26900 
6725 


48)9684,00($201,75 
96 
84 
48 
360 
336 


240 
240 


2.  If  6  men  can  dig  a  certain  ditch  in  40  days,  how  many 
days  would  30  men  be  employed  in  digging  it  ? 

As  the  answer  must  be  days. 


the  40  days  are  written  in  the 
third  place.  Then,  as  it  is 
evident  that  30  men  will  do 
the  same  work  in  a  shorter 
time  than  6  men,  it  is  plain 
that  the  fourth  term  must  be 


men 
30 


OPERATION. 

days 

40 

6 


men 
:    6 


days 
Ans, 


310)24|0  days, 
Ans.    8     days. 


less  than  the  third;  therefore,  30  men,  the  greater  of  the 
remaining  numbers,  is  taken  as  the  first  term.  Besides,  it 
is  plain  that  the  fourth  term  must  be  just  so  many  times  less 
than  40,  as  6  is  less  than  30. 

3.  If  25  yards  of  cloth  cost  £2  3^.  4cZ.,  what  will  5  yards 
cost  at  the  same  rate  ? 


Quest. — In  the  first  example  which  is  greater,  the  third  or  fourth  term  ? 
Which  number  must  then  be  in  the  first  term  ?  How  many  times  will  the 
fourth  term  be  greater  or  less  than  the  third  ? 


198 


RULE    OF    THREE. 


When  we  come  to 
divide  the  product  of 
the  second  and  third 
terms  by  the  first,  it  is 
found  the  £10  does  not 
contain  25.  We  then 
reduce  to  the  next  low- 
er denomination,  and 
divide  as  in  division  of 
denominate  numbers. 


yd, 
25 

OPERATION. 

yd.         £    s.    d. 

:    5    :  :    2     3     4 

5 

25)£10  16^.  8d 
20 

25)216(8^. 
200 


16 
12 


25)200(8(;. 
200 


Am 


Ans.  8s,  8d, 


4.  If  3cwt.  of  sugar  cost  £9  2^.  Od.,  what  will  4cwt.  3qr 
26lb,  cost  at  the  samfe  rate  ? 


4X7  =  28 


We  first  reduce  the 
first  and  second  terms 
to  pounds,  then  the 
third  term  to  pence. 
The  answer  comes  out 
in  pence,  and  is  af- 
terwards reduced  to 
pounds,  shillings,  and 
pence. 


4cwt,  3qr.  26/6. 
4 


2lS4:d, 

558 


£9   2s. 
20 


Od. 


1825. 
12 


2184 
:    Ans. 


17472 
10920 
10920 _ 

336)1218672(3627c?. 
1008 


2106 
2016 


12)3627 


20)  302^.  3d. 


907 
672 

2352 
2352 


£15  2^. 


Ans.   £15  2^.  3d. 


RULE    OF    THREE.  199 

PROOF. 

194.  The  product  of  the  two  means  is  equal  to  the  product 
of  the  extremes  (Art.  189).  Hence,  if  either  of  these  equal 
products  be  divided  by  one  of  the  mean  terms  the  quotient 
will  be  the  other.     Therefore, 

Divide  the  product  of  the  extremes  by  one  of  the  mean  terms, 
and  if  the  work  is  right  the  quotient  will  be  the  other  mean  term. 

EXAMPLES. 

1.  The  first  term  is  4,  the  second  8,  the  third  12,  and  the 
answer  24 :  is  the  answer  true  ? 


The  product  of  the  extremes 
is  96.     If  this  be  divided  by  8 
'the  quotient  is  12  ;  if  by  12  the 
quotient  is  8 :  hence,  the  an- 
swer is  right. 


OPERATION    OF    PROOF. 

24  ><:'  4  =  96 

8)96(12  ; 
or,    12)96(8 


APPLICATIONS. 

1.  If  8  hats  cost  $24,  what  will  110  cost  at  the  same  rate  ? 

2.  What  is  the  value  of  4cwt.  of  sugar  at  5d.  per  pound  ? 

3.  If  80  yards  of  cloth  cost  $340,  what  will  650  yards 
cost? 

4.  If  120  sheep  yield  330  pounds  of  wool,  how  many  pounds 
will  be  obtained  from  1200  ? 

5.  If  6  gallons  of  molasses  cost  $1,95,  what  will  6  hogs- 
heads cost  1 

6.  If  16  men  perform  a  piece  of  work  in  24  days,  how 
many  men  would  it  take  to  perform  the  work  in  12  days  ? 

7.  Suppose  a  cistern  has  two  pipes,  and  that  one  can  fill 
It  in  8i  hours,  the  other  in  4|- :  in  what  time  can  both  fill  it 
together  ? 

8.  If  a  man  travels  at  the  rate  of  630  miles  in  12  uays, 
how  far  will  he  travel  in  a  year,  supposing  him  not  to  trave] 
on  Sundays  1 


Quest. — 194.  How  do  you  prove  the  Rule  of  Three  ? 


200  RULE    OF    THREE. 

9.  If  2  yards  of  cloth  cost  $3,25,  what  will  be  the  cost  of 
3  pieces,  each  containing  25  yards  ? 

10.  If  30  barrels  of  flour  will  support  100  men  for  40  days, 
how  long  would  it  subsist  25  men  ? 

11.  If  30  barrels  of  flour  will  support  100  men  for  40  days, 
how  long  would  it  subsist  200  men  ? 

12.  If  50  persons  consume  600  bushels  of  wheat  in  a  year, 
how  much  will  they  consume  in  7  years  ? 

13.  What  will  be  the  cost  of  a  piece  of  silver  weighing 
73/^.  5oz.  I5pwt.j  at  5^.  9d.  per  ounce? 

14.  If  the  penny  loaf  weighs  8  ounces  when  the  bushel  of 
"wheat  costs. 7^.  3d.,  what  ought  it  to  weigh  when  the  wheat 
is  8^.  4d.  per  bushel  ? 

15.  If  one  acre  of  land  costs  £2  I5s.  4d.,  what  will  be  the 
cost  of  173^.  2R.  14P.  at  the  same  rate  ? 

16.  A  gentleman's  estate  is  worth  £4215  4^.  a  year :  what 
may  he  spend  per  day  and  yet  save  £1000  per  annum? 

17.  A  father  left  his  son  a  fortune,  \  of  which  he  ran 
through  in  8  months,  -^  of  the  remainder  lasted  him  12  months 
longer,  when  he  had  barely  £820  left :  what  sum  did  his 
father  leave  him  ? 

18.  There  are  1000  men  besieged  in  a  town  with  pro- 
visions for  5  weeks,  allowing  each  man  16  ounces  a  day. 
If  they  are  reinforced  by  500  more  and  no  relief  can  be 
aflJbrded  till  the  end  of  8  weeks,  how  many  ounces  must  be 
given  daily  to  each  man  ? 

19.  A  father  gave  -^  of  his  estate  to  one  son,  and  -^ 
of  the  remainder  to  another,  leaving  the  rest  to  his  widow. 
The  difference  of  the  children's  legacies  was  £514  6s.  8d. : 
what  was  the  widow's  portion  ? 

20.  If  14cwt.  2qr.  of  sugar  cost  $129,92,  what  will  be  the 
price  of  9cwt.  ? 

21.  The  clothing  of  a  regiment  of  foot  of  750  men  amounts 
to  £2831  5s.:  what  will  it  cost  to  clothe  a  body  of  10500 
men? 


RULE    OF    THREE.  201 

22.  How  many  yards  of  carpeting,  that  is  3  feet  wide,  will 
cover  a  floor  that  is  40  feet  long  and  27  feet  broad  ? 

23.  After  laying  out  ^  of  my  money,  and  ^  of  the  remain- 
der, I  had  114  guineas  left:  how  much  had  I  at  first? 

24.  A  reservoir  has  three  pipes,  the  first  can  fill  it  in  24 
days,  the  second  in  22  days,  and  the  third  can  empty  it  in 
28  days  :  in  what  time  will  it  be  filled  if  they  are  all  running 
together  ? 

25.  If  the  freight  of  80  tierces  of  sugar,  each  weighing 
S^cwt.,  150  miles,  cost  ^84,  what  must  be  paid  for  the  freight 
of  SOhhd.  of  sugar,  each  weighing  12cz^^.,  50  miles  ? 

26.  If  1500  men  require  45000  rations  of  bread  for  a 
month,  how  many  rations  will  a  garrison  of  3600  men  re- 
quire ? 

27.  The  quick  step  in  marching  is  2  paces  per  second,  at 
28  inches  each  :  at  what  rate  is  that  per  hour,  and  how  long 
will  a  troop  be  in  reaching  a  place  60  miles  distant,  allowing 
a  halt  of  an  hour  and  a  half  for  refreshment  ? 

28.  Two  persons  A  and  B  are  on  the  opposite  sides  of  a 
wood  which  is  536  yards  in  circumference ;  they  begin  to 
travel  in  the  same  direction  at  the  same  moment ;  A  goes  at 
the  rate  of  11  yards  per  minute,  and  B  at  the  rate  of  34  yards 
in  3  minutes  :  how  many  times  must  the  quicker  one  go  round 
the  wood  before  he  overtakes  the  slower  ? 

29.  Two  men  and  a  boy  were  engaged  to  do  a  piece  of 
work,  one  of  the  men  could  do  it  in  10  days,  the  other  in  16 
days,  and  the  boy  could  do  it  in  20  days  :  how  long  would  it 
take  the  three  together  to  do  it  ? 

30.  A  certain  amount  of  provisions  will  subsist  an  army 
of  9000  men  for  90  days.  If  the  army  be  increased  by  6000, 
how  long  will  the  same  provisions  subsist  it  ? 

31.  Four  thousand  soldiers  were  supplied  with  bread  for 
24  weeks,  each  man  to  receive  lAoz.  per  day;  but  by  some 
accident  210  barrels  containing  2001b.  each  were  spoiled: 
what  must  each  man  receive  in  order  that  the  remainder  may 
last  the  same  time  ? 

9* 


202  RULE    OF    THREE. 

32.  Let  us  suppose  the  4000  soldiers  having  one-fourteenth 
of  their  bread  spoiled,  to  be  put  on  an  allowance  of  l3oz.  of 
bread  per  day  for  24  weeks :  required  the  weight  of  their 
bread,  good  and  spoiled,  and  the  amount  spoiled. 

33.  If  56  yards  of  cloth  cost  40  guineas,  how  many  ells 
Flemish  can  be  bought  for  £1135  10^.? 

34.  If  a  pack  of  wool  weighs  2cwt.  2qr.  I4lb,,  what  is  it 
worth  at  17^.  6d.  per  tod  1 

35.  A  merchant  bought  a  quantity  of  broadcloth  and  baize 
for  £124  ;  there  was  117i  yards  of  broadcloth  at  I7s.  9d.  per 
yard;  for  every  5  yards  of  broadcloth  he  had  1^-  yards  of 
baize ;  how  many  yards  of  baize  did  he  buy,  and  what  did 
it  cost  him  per  yard  1 

36.  If  J  of  a  pole  stands  in  the  mud,  1  foot  in  the  water, 
and  I"  in  the  air,  or  above  the  water,  what  is  the  length  of 
the  pole  ? 

37.  A  bankrupt's  effects  amount  to  lOOOi  guineas.  His 
debts  amount  to  £2547  14^.  9d.  :  what  will  his  creditors  re- 
ceive in  the  pound  ^ 

38.  If  12  dozen  copies  of  a  certain  book  cost  $54,72,  what 
will  297  copies  cost  at  the  same  rate  ? 

39.  Suppose  4000  soldiers  after  losing  210  barrels  of  bread, 
each  containing  2001b.,  were  to  subsist  on  13oz.  each  a  day 
for  24  weeks  ;  had  none  been  lost  they  might  have  received 
I4:0z.  a  day :  what  was  the  whole  weight,  and  how  much  did 
they  receive  ? 

40.  Let  us  now  suppose  4000  soldiers  to  lose  one-four- 
teenth of  their  bread,  then  to  receive  13o;^.  each  a  day  for  24 
weeks :  what  was  the  whole  weight  of  their  bread  including 
the  lost,  and  how  much  would  each  have  received  per  day 
had  none  been  spoiled  ? 

41.  Provisions  in  a  garrison  were  sufficient  for  1800  men 
for  12  months  ;  but  at  the  end  of  3  months  it  was  reinforced 
by  600  men,  and  4  months  after  that  a  second  reinforcement 
of  400  men  was  sent  in.      How  long  did  the  provisions  last? 


RULE    OF    THREE    BY    ANALYSIS.  203 


RULE  OF  THREE  BY  ANALYSIS. 

195.  The  solution  of  questions  in  the  Rule  of  Three  by- 
analysis  consists  in  finding  the  ratio  of  two  of  the  given 
terms,  and  multiplying  this  ratio  by  the  other  term. 

The  ratio  of  two  of  the  terms  will  generally  express  the 
\ralue  or  cost  of  a  single  thing. 

EXAMPLES. 

1 .  If  3  barrels  of  flour  cost  $24,  what  will  7  barrels  cost  ^ 

OPERATION 

3)24 
8 


By  dividing  the  $24  by  3  we  get  the 
cost  of  1  barrel.  For,  if  $24  will  buy  3 
barrels,  it  is  plain  that  ^  of  it  will  buy  1 
barrel.  This,  multiplied  by  7,  gives  $56 
the  cost  of  7  barrels. 


8  X  7  =  56 
Ans.  $56. 

2.  If  in  20  days  a  man  travels  58  miles,  how  far  will  he 
travel  in  30  days  ? 

3.  If  6  men  consume  1  barrel  of  flour  in  30  days,  how 
much  would  48  men  consume  ? 


It  is  evident  that  ^  of  a  barrel  would  be 


OPERATION. 

^  X  48  z^  8. 


the  amount  consumed  by  1  man :  hence, 

48  time's  ^  is  the  amount  consumed  by  48  Ans7^8, 

men.  ' 

4.  If  2  barrels  of  flour  cost  $13,  what  will  12  barrels  cost  ? 

5.  If  I  walk  168  miles  in  6  days,  how  far  should  I  walk  at 
the  same  rate  in  18? 

6.  If  8/^.  of  sugar  cost  $1,28,  how  much  will  13Z^>.  cost? 
What  is  16  X  13? 

7.  If  |-  of  a  piece  of  cloth  cost  $8,25,  what  will  |  cost  ? 

8.  If  300  barrels  of  flour  cost  $570,  what  will  200  cost  ? 
What  is  f  X  570  ? 

9.  If  "I"  of  a  barrel  of  cider  cost  ^j  of  a  dollar,  what  will  | 
cost  ?     What  is  fl  X  y®T  ? 

Quest. — 195.  In  what  does  the  solution  of  questions  by  analysis  consist ' 
What  does  the  ratio  of  the  two  terms  express  ?  If  this  ratio  be  multiplie{j 
by  the  other  term,  what  will  be  the  product  ? 


204  RULE  OP  THREE  BY  ANALYSIS. 

10  If  90  bushels  of  oats  will  feed  40  horses  for  6  days, 
how  long  would  450  bushels  feed  them  ? 

11.  If  5  oxen,  or  7  colts,  eat  up  a  certain  quantity  of  grass 
in  87  days,  in  what  time  will  2  oxen  and  3  colts  eat  up  the 
same*  quantity  of  grass  ? 

12.  A  person's  income  is  £146  per  annum  :  how  much  is 
that  each  day  ? 

13.  If  3  paces  of  common  steps  be  equal  to  2  yards,  how 
many  yards  will  160  paces  make  ? 

14.  A  certain  work  can  be  done  in  12  days  by  working  4 
hours  each  day  :  how  long  would  it  require  to  do  the  work  by 
working  9  hours  a  day  ? 

15.  A  pasture  of  a  certain  extent  having  supplied  a  body 
of  ^orse,  consisting  of  3000,  with  forage  for  18  days,  how 
many  days  would  the  same  pasture  have  supplied  a  body  of 
2000  horse  ? 

16.  The  governor  of  a  besieged  city  has  provisions  for  54 
days,  at  the  rate  of  2lh.  of  bread  per  day,  but  is  desirous  of 
prolonging  the  siege  to  80  days,  in  expectation  of  succor :  in 
that  case  what  must  be  the  allowance  of  bread  per  day  ? 

17.  If  a  person  pa^s  half  a  guinea  a  week  for  his  board, 
how  long  can  he  be  boarded  for  £21  ? 

18.  If  a  person  drinks  80  bottles  of  wine  per  month,  when 
it  costs  2s.  per  bottle,  how  much  can  he  drink,  without  in- 
creasing the  expense,  when  it  costs  2^.  Qd.  per  bottle  ? 

19.  How  long  will  a  person  be  in  saving  £100,  if  he  saves 
1^.  Qd.  per  week? 

20.  A  merchant  bought  21  pieces  of  cloth,  each  containing 
41yards,lbr  which  he  paid  $1260  ;  he  sold  the  cloth  at  $1,75 
per  yard  :  did  he  gain  or  lose  by  the  bargain  ? 

21.  A  grocer  bought  a  puncheon  of  rum  for  £41  145.  6J., 
to  which  he  added  as  much  water  as  reduced  its  cost  to  bs, 
6d.  per  gallon ;  how  much  water  did  he  put  in  ? 

22.  If  one  pound  of  tea  be  equal  in  value  to  50  oranges 
and  70  oranges  be  worth  84  lemons,  what  is  the  value  of  a 
pound  of  tea  when  a  lemon  is  worth  two  cents  ? 


RULE  OF  THREE  BY  CANCELLING.        205 


RULE  OF  THREE  BY  CANCELLING. 

196.  The  cancelling  process  may  be  applied  to  all  ques 
tions  in  the  Rule  of  Three,  where  the  second  or  third^  terms 
have  a  factor  common  to  the  first.  Let  the  second  and  third 
terms  be  written  above  the  line,  with  the  sign  of  multiplica- 
tion between  them,  and  the  first  term  below  it,  and  then  can- 
cel the  common  factors. 

EXAMPLES. 

1.  If  48  yards  of  cloth  cost  $67,25,  what  will  144  yards 
cost  ? 

OPERATION. 


The  process  here  is  obvious, 
being  entirely  similar  to  that 
explained  in  Art.  191. 


.3 
67,25  X  AAA 

48 


:  $201,75. 


2.  If  25  yards  of  cloth  cost  £2  3^.  Ad.,  what  will  5  yards 
cost? 

3.  If  24  hats  cost  $120,  how  much  will  80  cost? 

4.  If  90  barrels  of  flour  will  subsist  100  .  men  for  120 
days,  how  long  will  it  subsist  75  ? 

5.  If  60  sheep  yield  180/6.  of  wool,  how  many  pounds  will 
be  obtained  from  900  ? 

6.  If  a  man  travel  210  miles  in  6  days,  how  far  will  he 
travel  in  120  days  ? 

7.  If  the  freight  of  40  tierces  of  sugar,  each  weighing 
^\cwt.,  150  miles,  cost  $42,  what  must  be  paid  for  the  freight 
of  10  hogsheads,  each  weighing  I2cwt.,  50  miles  ? 

8.  A  certain  amount  of  provisions  will  subsist  an  army  of 
9000  men  for  90  days  :  if  the  army  be  increased  by  4500 
how  long  would  the  same  provisions  subsist  it  ? 

9.  If  50  persons  consume  600  bushels  of  wheat  in  one 
jrear,  how  much  will  278  persons  consume  in  7  years  ? 

10.  If  3  yards  of  cloth  cost  18^.,  what  will  24  yards  cost  ? 

Quest. — 196.  To  what  questions  may  the  cancelling  process  be  applied* 
flow  are  the  numbers  written?     What  factors  do  you  cancel? 


206  RULE    OF    THREE    BY    CANCELLING. 

11.  If  112  pounds  of  sugar  cost  56^.,  what  will  1  pound 
cost? 

12.  If  4  men  can  do  a  piece  of  work  in  80  days,  how  many 
days  of  the  same  length  will  16  men  require  to  do  the  same 
work  ?• 

13.  If  21  pioneers  make  a  trench  in  18  days,  how  many 
days  of  the  same  length  will  7  men  require  to  make  a  similar 
trench  ? 

14.  If  a  field  of  grass  be  mowed  by  10  men  in  12  days,  in 
how  many  days  would  it  be  mowed  by  20  men? 

15.  A  certain  piece  of  grass  was  to  be  mowed  by  20  men 
in  6  days  ;  an  extraordinary  occasion  calls  off  half  the  work- 
men.   It  is  required  to  find  in  what  time  the  rest  will  finish  it. 

16.  If  the  pejiny  loaf  weighs  5oz.  when  flour  is  2^.  a 
peck,  what  should  it  weigh  when  flour  is  sold  for  2^.  6d,  a 
peck? 

17.  Provisions  in  a  garrison  are  found  sufficient  to  last 
1800  soldiers  for  three  months ;  but  a  reinforcement  being 
wanted  that  the  provisions  may  last  for  one  month  only,  what 
number  of  soldiers  must  be  added  to  the  garrison  ? 

18.  If  3i/d.  2qr,  of  cloth  of  li/d.  Sqr.  wide  will  make  a  suit 
of  clothes,  how  many  yards  of  stuff  of  Sqr.  wide  will  make  a 
suit  for  the  same  person  ? 

19.  If  I  lend  my  friend  £200  for  12  months  on  condition 
of  his  returning  the  favor,  how  long  ought  he  to  lend  me  £150 
to  requite  my  kindness  ? 

20.  If  an  acre  be  220  yards  long,  the  breadth  will  be  22 
yards ;  but  if  the  breadth  of  an  acre  be  40  yards,  what  then 
will  be  the  length  ? 

21.  How  many  pounds  of  sugar  at  12^.  per  pound  are 
equal  in  value  to  24:1b.  of  tea,  worth  9^.  6d.  per  pound  ? 

22.  A  tax  of  £225  10^.  was  laid  upon  four  villages  A,  B, 
C,  and  D  ;  it  had  been  the  custom  with  these  villages,  that 
whenever  any  taxes  were  to  be  levied,  as  often  as  A,  B,  and 
C  paid  each  36?.,  D  paid  only  2c?. :  how  much  did  each  village 
pay? 


OPERATION. 

• 

f  :  2i  :  :  3,20  : 

Ans. 

H 

6,40 

by 

multiplying  by  ^      1 ,60 

8,00 

8,00 

-.  1  :=:  8,00   X  f  = 

zir  $21,33+. 

=  ^v" 

EXAMPLES    INVOLVING    FRACTIONS.  207 

EXAMPLES    INVOLVING    FRACTIONS. 

1.  If  I  of  a  yard  of  cloth  cost  $3,20,  what  will  2^  yards 
cost? 

We  state  the  ques- 
tion exactly  as  in 
whole  numbers.  In 
multiplying  the  sec- 
ond and  third  terms 
together,  we  observe 
the  rules  for  multi- 
plying fractions,  and 
in  dividing  by  the  first 
term,  the  rules  for  division.  Thus,  in  this  example,  we  in- 
vert the  terms  of  the  divisor  and  multiply.    * 

2.  If  ^oz.  cost  £j^j  what  will  l^oz,  cost?       Ans.  

3.  If  ^3_  of  a  ship  cost  £273  3^.  6d.y  what  will  ^  of  her 
cost? 

4.  If  375  yards  of  cloth  cost  £164  9d,  what  will  257^  yards 
cost  ? 

5.  If  14  yards  of  cloth  can  be  bought  for  10  guineas,  how 
many  ells  Flemish  can  be  bought  for  £283.875  ? 

6.  If  }^oz.  of  plate  cost  10^.  ll^d.,  what  will  a  service 
weighing  327.61875o2r.  cost? 

7.  If  14fZ6.  of  sugar  cost  $1|,  what  will  12Z5.  cost? 

8.  If  I  of  a  yard  of  cloth  cost  $1|-,  what  will  7i  yards 
cost? 

9.  If  2  men  can  do  125  rods  of  ditching  in  65  days,  in  how 
many  days  can  18  men  do  242^^  rods  ? 

10.  If  a  wedge  of  gold  weighing  ll^lb.  troy,  be  worth 
£679^,  what  is  the  value  of  Ij^qr.  of  that  gold  ? 

11.  If  the  carriage  of  2.5  tons  of  goods  2.9  miles  cojt 
0.75  guinea,  what  is  that  per  cwt.  for  a  mile  ? 

12.  If  ^cwt.  of  tobacco  cost  £4  18^.,  how  much  may  bo 
bought  for  £7  ? 


208  EXAMPLES    INVOLVING    FRACTIONS 

13.  If  14i  yards  of  cloth  cost  $19i  how  much  will  39| 
yards  cost  ? 

14.  If  .3  of  a  house  cost  $201.5,  what  would  .95  cost? 

15.  A  man  receives  |  of  his  income,  and  finds  it  equal  to 
$3724.16  :  how  much  is  his  whole  income  ? 

16.  If  3.5  yards  of  cloth  cost  £2  Us.  3d.,  what  will  27.75 
yards  cost  at  the  same  rate  1 

17.  If  12  men  and  a  boy  can  perform  a  piece  of  work  in 
100|^  days,  the  boy  doing  i  as  much  work  as  one  man,  in  how 
many  days  will  20  men  perform  the  same  ? 

18.  A  mercer  bought  10^  pieces  of  silk,  each  containing 
241  yards  ;  he  paid  6s.  ^d.  per  yard  :  what  does  the  whole 
come  to  ? 

19.  If  4lb.  oTbeef  cost  -|  of  a  dollar,  what  will  60lb.  cost  ? 

20.  If  a  traveller  perform  a  journey  in  35.5  days  when  the 
days  are  13.625  hours  long;  in  how  many  days  of  11.9 
hours  would  he  perform  the  same  journey  ? 

21.  If  5400  bricks  be  required  to  pave  a  yard,  when  the 
bricks  are  .5  foot  long  and  .25  broad,  how  many  will  be  re- 
quired of  .75  foot  long  and  i  foot  broad  ? 

22.  A  man  with  his  family,  consisting  of  5  persons,  usually 
drink  7.8  gallons  of  beer  in  a  week  :  how  much  would  they 
drink  in  23.5  weeks,  if  the  family  was  to  be  increased  by  3 
persons  ? 

23.  If  248  men  in  601  hours  dig  a  trench  containing 
139241  solid  yards  of  earth,  how  long  would  it  take  the 
same  number  of  men  to  dig  a  similar  trench  containing  26460 
solid  yards  of  earth  ? 

24.  The  earth  turns  round  on  its  axis  from  west  to  east  in 
23  hours  56  minutes,  and  the  circumference  of  every  circle 
on  its  surface  is  supposed  to  be  divided  into  360  degrees. 
At  the  equator  a  degree  is  69.07  miles  ;  at  Madras,  Barba- 
does,  &c.,  67.21  miles  ;  at  Madrid,  Philadelphia,  &c.,  52.85 
miles  ;  and  at  Petersburg,  34.53  miles.  How  many  miles 
per  hour  are  the  inhabitants  in  each  of  these  places  carried 
from  west  to  east  by  the  revolution  of  the  earth  on  its  axis  ? 


QUESTIONS    REQUIRING    TWO    STATEMENTS. 


209 


OF  QUESTIONS  REQUIRING  TWO  STATEMENTS. 

197.  The  answer  to  each  of  the  questions  heretofore  con- 
sidered, has  been  found  by  a  single  statement.  Questions, 
however,  frequently  occur  in  which  two  or  more  statements 
will  be  necessary,  if  the* question  be  resolved  by  the  princi- 
ples above  explained. 


1st  operation. 
persons,    persons.  $ 

BY  CANCELLING. 

2      :      5     :   :     300 
5 


Ans, 


EXAMPLES. 

1.  If  a  family  of  6  persons  expend  $300  in  8  months,  how 
much  will  serve  a  family  of  15  persons  for  20  months  ? 

First  question.  If 
$300  will  support  a 
family  of  6  persons 
for  8  months,  how 
many  dollars  will  sup- 
port 15  persons  for 
the  same  time  ? 

Second  question.  If 
$750  will  support  a 
family  of  15  persons 
for  8  months,  how 
much  will  serve  them 
for  20  months  ? 

2.  If  16  men  build  18  feet  of  wall  in  12  days,  how  many 
men  must  be  employed  to  build  72  feet  in  8  days,  working  at 
the  same  rate  ? 

The  first  question 
is,  how  long  would  it 
take  the  16  men  to 
build  the  72  feet  of 
wall  ? 

It  is  evident  that  18  feet  of  wall,  is  to  72  feet,  as  12  days, 


months. 
2      : 


2)1500 

Ans 

.     $750. 

2d    OPERATION. 
onths.                 $ 

5      :    :      750 
5 

2)3750 

Ans. 

$1875 

Ans, 


OPERATION. 

feet, 

1 

feet,            days, 
:      4      :    :      12 
4 

days, 
Ans, 

48  days. 


Quest. — 197.  How  many  statements  have  been  necessary  in  the  ques- 
tions heretofore  considered?     What  other  questions  frequently  occur? 


210 


double  rule  of  three. 


the  time  necessary  to  build  18  feet,  is  to  48  days,  the  time 
necessary  to  build  72  feet. 

The   second  ques- 
tion is,  if  16  men  can        days, 
build  72  feet  of  wall  8 

in  48  days,  how  many 
men  are  necessary  to 
build  it  in  8  days  ? 

Make  16  men  the 
third  term.  Then  as 
the  same  work  is  to 
be  done  in  less  time,  more  men  will  be  necessary ;  therefore, 
the  fourth  term  will  be  greater  than  the  third,  and  hence  8 
days  are  placed  in  the  first  term  (Art.  193). 

3.  If  a  man  travel  217  miles  in  7  days,  travelling  6  hours 
a  day,  how  far  would  he  travel  in  9  days,  if  he  travelled  1 1 
hours  a  day? 


operation. 

days. 
48      : 

men, 
:      16 

48 

' 

men 
Ans 

128 
64 

Ans 

8)768 
96 

men 

1st  operation. 

days.          miles. 

9     :  :    217    ; 

9 

7)1953 

279 

miles. 
279 

hours. 
6 

2d  operation. 
hours.          miles.       miles. 
11     :  :    279    :    511| 
11 
6)3069 
511f 

Ans.  51 1|-  miles 

DOUBLE  RULE  OF  THREE. 

198.  The  last  three  questions,  and  all  similar  ones  in- 
volving five,  seven,  or  even  nine  terms,  have  generally  been 
classed  under  a  separate  rule,  called  the  Double  Rule  of 
Three,  or  Compound  Proportion.  They  may  be  thus 
stated  and  resolved : 

I.  Make  the  first  statement  as  though  the  question  were  to  he 
solved  hy  two  or  more  statements  hy  the  Single  Rule  of  Three, 
and  suppose  the  fourth  term  to  be  found. 

Quest. — 198.  Under  what  rule  have  questions  similar  to  the  last  three 
been  classed  ?     How  may  they  be  stated  and  resolved  ?     Give  the  rule. 


DOUBLE  RULE  OF  THREE.  211 

II.  If  it  is  of  the  same  name  with  the  answer  sought,  mark 
its  place  blank  under  the  third  term ;  if  not,  mark  its  place  un- 
der the  second  term,  and  in  either  case  arrange  the  two  remain- 
ing terms  as  though  it  were  a  question  in  the  Single  Rule  of 
Three.  If  there  are  more  than  five  terms  in  the  question,  sup- 
pose the  fourth  term  of  the  second  proportion  to  he  found,  and 
make  the  third  statement  in  the  same  manner  as  the  second  was 
made, 

III.  Then  multiply  the  second  and  third  terms  together,  and 
divide  their  product  by  the  product  of  the  first  terms,  and  the 
quotient  will  be  the  answer  sought. 

EXAMPLES. 

Let  us  first  resolve  each  of  the  last  three  questions  by  this 
rule. 

1.  If  a  family  of  6  persons  expend  $300  in  8  months,  how 
much  will  serve  a  family  of  15  persons  for  20  months  ^^ 

OPERATION. 

persons,  persons,  $ 

6      :       15      :  :        300         :     1^^  answer 
months,   months, 

8      :      20      :  :      1^^  ans,     :     true  answer 


25. 

5        ^jS 

15xX/xX|// 


=  15  X  5  X  25  =  $1875. 


^X^ 

Having  made  the  first  statement,  we  see  that  the  fourth 
term  is  of  the  same  name  with  the  answer  sought,  and  that  if 
this  term  be  placed  in  the  second  proportion,  the  true  answer 
will  be  found.  But  since  the  first  answer  is  equal  to  the 
product  of  the  second  and  third  terms  divided  by  the  first,  it 
is  plain  that  the  true  answer  will  always  be  equal  to  the  con- 
tinued product  of  the  second  and  third  terms  divided  by  the 
product  of  the  first  terms,  and  similarly  when  there  are  more 
than  five  terms.  In  the  operation  we  first  cancel  the  6  in 
the  300,  then  the  4  from  20,  and  then  the  2  from  the  50  over 
the  300. 


212  DOUBLE  RULE  OF  THREE. 

2.  If  16  men  build  18  feet  of  wall  in  12  days,  how  many 
men  must  be  employed  to  build  72  feet  in  8  days,  working  at 
the  same  rate  ? 

OPERATION. 

feet,       feet*  days, 

18  :     72     :  :     12     :     1^^  answer, 
days,      days.  men, 

8  :     —     :  :     16     :     true  answer, 
4                    2 

Then,      ^|-i?4^=4xl2x2  =  96^n.. 

3.  If  a  man  travel  217  miles  in  7  days,  travelling  6  hours 
a  day,  how  far  would  he  travel  in  9  days,  if  he  travelled  11 
hours  a  day  ? 


OPERATION. 

ays. 

days, 
3 

miles. 

miles. 

7     : 

^ 

:  :     217     : 

\st  answer 

^    : 

11 

true  answet. 

4.  If  4  compositors,  in  16  days  of  12  hours  long,  can  com- 
pose 14  sheets  of  24  pages  each  sheet,  44  lines  in  a  page, 
and  40  letters  in  a  line  ;  in  how  many  days  of  10  hours  long 
will  9  compositors  compose  a  volume  consisting  of  30  sheets, 
16  pages  in  a  sheet,  48  lines  in  a  page,  and  45  letters  in  a 
line? 

The  number  of  letters  set  by  the  first  compositors  is  ex- 
pressed by  14  X  24  X  44  X  40 ;  and  the  letters  to  be  set 
by  the  second  by  30  x  16  X  48  x  45. 


OPERATION. 

com. 

com. 

days. 

Ans,' 

9  : 
ours. 

4 

hours. 

:  :   16  : 

1st  answer. 

10  : 

12 

:  :  —  : 

2d  answer. 

14x24x44x40  :  30x16x48x45  :  :  —  :  true  answer. 


DOUBLE    RULE   OF   THREE.  213 

4xl2xl6x30xl6x48x45_4x3xl6x3x2_ 
9x10x14x24x44x40      ~  7x11  "~ 

=  -^  =  14^4-  days  =  Ans. 

Let  us  now  analyze  this  statement.  Had  the  compositors 
worked  the  same  number  of  hours  per  day,  and  had  the  same 
work  to  do,  the  first  would  have  been  the  true  answer ;  and 
the  second  would  have  been  the  true  answer  had  the  time 
only  been  different  and  the  work  to  be  done  been  the  same. 
The  third  proportion  accounts  for  the  inequality  of  the  work 
done,  and  gives  the  answer  under  all  the  suppositions.  It  is 
evident  the  same  answer  would  have  been  obtained,  had  the 
first  answer  been  substituted  in  the  second  proportion,  and 
the  second  answer  in  the  third  proportion.  Hence,  the  rea- 
son of  the  rule  is  obvious. 

5.  If  a  pasture  of  16  acres  will  feed  6  horses  for  4  months, 
how  many  acres  will  feed  12  horses  for  9  months  ? 

6.  If  25  persons  consume  300  bushels  of  corn  in  1  year, 
how  much  will  139  persons  consume  in  7  years  at  the  same 
rate  ? 

7.  If  32  men  build  a  wall  36  feet  long,  8  feet  high,  and  4 
feet  wide  in  4  days  ;  in  what  time  will  48  men  build  a  wall 
864  feet  long,  6  feet  high,  and  3  feet  wide  ? 

8.  If  a  regiment  of  1878  soldiers  consume  702  quarters  of 
wheat  in  336  days,  how  many  quarters  will  an  army  of  22536 
soldiers  consume  in  112  days? 

9.  If  12  tailors  in  7  days  can  finish  13  suits  of  clothes, 
how  many  tailors  in  19  days  of  the  same  length,  can  finish 
the  clothes  of  a  regiment  of  soldiers  consisting  of  494  men  ? 

10.  An  ordinary  of  100  men  drank  £20  worth  of  wine  at 
25.  Qd,  per  bottle  ;  how  many  men,  at  the  same  rate  of  drink- 
ing, will  £7  worth  suffice,  when  wine  is  rated  at  1^.  9d.  per 
bottle  ? 

11.  If  60  bushels  of  oats  will  serve  24  horses  for  40  days, 
how  long  will  30  bushels  serve  48  horses  at  the  same  rate  ? 

12.  If  a  garrison  of  3600  men,  in  35  days,  at  2^oz,  per 


214  DOUBLE    RULE    OF    THREE. 

day  each  man,  eat  a  certain  quantity  of  bread,  how  many 
men  in  45  days,  at  the  rate  of  lioz.  per  day  each  man,  will 
eat  double  the  quantity  ? 

13.  A  garrison  of  3600  men  has  just  bread  enough  to  allow 
24:Oz.  a  day  to  each  man  for  35  days ;  but  a  siege  coming 
on,  the  garrison  was  reinforced  to  the  number  of  4800  men. 
How  many  ounces  of  bread  a  day  must  each  man  be  allowed, 
to  hold  out  45  days  against  the  enemy  1 

14.  If  336  men,  in  5  days  of  10  hours  each,  dig  a  trench 
of  5  degrees  of  hardness,  70  yards  long,  3  wide,  and  2  deep, 
what  length  of  trench  of  6  degrees  of  hardness,  5  yards  wide, 
and  3  deep,  may  be  dug  by  240  men  in  9  days  of  12  hours 
each? 

15.  If  12  pieces  of  cannon,  eighteen-pounders,  can  batter 
down  a  castle  in  an  hour,  in  what  time  would  nine  twenty- 
four-pounders  batter  down  the  same  castle,  both  pieces  of  can- 
non being  fired  the  same  number  of  times,  and  their  balls 
flying  with  the  same  degree  of  velocity? 

16.  If  15  weavers  by  working  10  hours  a  day  for  10  days, 
can  make  250  yards  of  cloth,  how  many  must  work  9  hours 
a  day  for  1 5  days,  to  make  607^  yards  ? 

17.  If  £3i  be  the  wages  of  13  men  for  7^  days,  what« 
will  be  the  wages  of  20  men  for  15^  days  ? 

18.  If  a  footman  travel  294  miles  in  7f  days,  of  12J  hours 
long,  in  how  many  days,  of  lOf  hours  long  each,  will  he 
travel  147^  miles? 

19.  Bought  5000  planks,  of  15  feet  long  and  2^  inches 
thick ;  how  many  planks  are  they  equivalent  to,  of  12^  feet 
long  and  If  inches  thick? 

20.  If  248  men,  in  5^  days  of  11  hours  each,  dig  a  trench 
of  7  degrees  of  hardness,  232^  yards  long,  3f  wide,  and  2^ 
deep ;  in  how  many  days,  of  9  hours  long,  will  24  men  dig  a 
trench  of  4  degrees  of  hardness,  337J  yards  long,  5f  wide, 
and  3^  deep  ? 


PRACTICE. 


215* 


PRACTICE. 


199.  Practice  is  an  easy  and  concise  method  of  apply- 
ing the  rules  of  arithmetic  to  questions  which  occur  in  trade 
and  business.  It  is  only  a  contraction  of  the  Rule  of 
Three  when  the  first  term  is  unity. 

For  example,  if  1  yard  of  cloth  cost  half  a  dollar,  what  will 
60  yards  cost?  This  is  a  question  which  may  be  answered 
by  the  rule  called  Practice.     The  cost  is  obviously  $30. 

200.  One  number  is  said  to  be  an  aliquot  part  of  another, 
when  it  forms  an  exact  part  of  it :  that  is,  when  it  is  con- 
tained in  that  other  an  exact  number  of  times.  Hence,  an 
aliquot  part  is  an  exact  or  even  part. 

For  example,  25  cents  is  an  aliquot  part  of  a  dollar.  It  is 
an  exact  fourth  part,  and  is  contained  in  the  dollar  four  times. 
So  also,  2  months,  3  months,  4  months,  and  6  months,  are  all 
aliquot  parts  of  a  year. 


TABLE    OF    ALIQUOT   PARTS. 


Cts. 

Farts 
of$l. 

Mo. 

Parts  of  a 
year. 

Days. 

Parts  of 
Imo. 

Parts  of  jCI. 

Parts  of 
1  shilling. 

50  = 
33i= 
25  = 
20  = 
12i= 

5  = 

2V 

6= 
4= 
3= 
2= 
1= 

i 

i 

or  ^1  of 
3  mo. 

15  = 
10  = 

^= 

6  = 
5  = 
3  = 

i 

1 

i 
i 

105.       ==  I 

6s.  8d.=  i 
5s.        =i 
4s.        =i 
3s.  4d.=  i 
2s.  6d.=:  ^ 
Is.  8d.=^ 

6  ^.  =  1 
^d.=i 
3  d.=i 
2  d.  =  i 

nj.  =  i 
1  d.=^ 

Quest. — 199.  What  is  Practice  ?  If  one  yard  of  cloth  cost  $8,  what 
will  half  a  yard  cost  ?  What  will  one  quarter  of  a  yard  cost  ?  200.  When 
is  one  number  said  to  be  an  aliquot  part  of  another  ?  What  is  an  aliquot 
part?  What  are  the  aliquot  parts  of  a  dollar  expressed  in  the  table? 
What  the  aliquot  parts  of  a  year  ?  What  the  aliquot  parts  of  a  month  ? 
What  the  aliquot  parts  of  a  pound  ?  What  are  the  aliquot  parts  of  a 
Bhilling  ? 


216 


PRACTICE. 


),75,  or  I  of 


OPERATION. 


EXAMPLES. 

1 .  What  is  the  cost  of  376  yards  of  cloth  at 
a  dollar  per  yard  ? 

Had  the  cloth  cost  $1 
per  yard,  the  cost  of  the 
376    yards    would    have 

been  $376.     Had  it  cost  188       cost  at  50cts. 

50cts.  per  yard,  the  cost        25      1  _94       cost  at  25cts. 

^     ^^!      "75"    I   $282       cost  3,t  Moll 
or    $188:    had   it * 

been  25cts,  per  yard,  the  cost  would  have  been  ^  of  $376, 

or  $94 ;  but  the  price  being  75cts»  per  yard,  the  cost  is 

188  +  94  =  i 


would    have    been   ^ 


cts, 
50 

25 

i 

$ 
376 

188 
94 

75 

3 

4 

$282 

2.  What  is  the  cost  of  196  yards  of  cotton,  at  9c?.  per 
yard? 

I96yd,  at  6d.  or  ^s.  =    9Ss. 
Iddyd.  at  3d.  or  ^s.  =    49^. 
Therefore,     I96yd.  at  9d,  or  ^s.  =  147^.  =  £7  7s.  Ans. 


3.  What  is  the  cost  of 
4715  yards  of  tape,  at  ^d. 
per  yard  ? 

id.  -  -  4)4715 

I2)n78^d.  =  cost. 
20)98s.2}d. 
Ans.  =  £4  18;?.  2j(^. 


4.  What  is  the  cost  of  425 
yards  at  1  penny  per  yard  1 

ld.=-^s.  -  12)425 

20)3E's.  5d. 
Ans.  £1  I5s.  5d, 


5.  What  will  be  the  cost 
of  354  yards  at  l\  per  yard  ? 
s.  -    12)354 

4)29^.    6d. 


ld.=^. 


\d.   - 


7^. 


4^d. 


cost     SGs.'lO^d. 
=r  £1  I6s.  lO^d. 


6.  What  will  be  the  cost 
of  4756  yards  of  cotton  shirt- 
ing, at  121  cents  per  yard  ? 

12^c^;y.=:iof  1$.    8)4756 
594^ 
Ans.  $594,50. 


PRACTICE. 


217 


7.  What  will  be  the  cost 
of  3754   pairs  of  gloves,  at 
2^.  6d.  per  pair  ? 
2^.  6d.z=}£,  -  -  8)37^54 
469f 
Ans.  £469  5^. 


8.  What  will  be  the  cost 
of  5320  bushels  of  wheat,  at 
3^.  6d.  per  bushel  ? 

5320 

3 

at  3^. 15960 

ati^.    -  -  -  -     2660 

at  3^.  6d.    -  -  18620^. 
Ans,  £9310 


9.  What  will  be  the  cost 
of  435  yards  of  cloth,  at  £2 
7*.  per  yard  ? 


Cost  at  £2     - 
5s,=zi  of£. 


435 
2_ 

£870 
108  15^. 
43  10^. 


Total  cost  £1022    56'. 


10.  What  will  be  the  cost 
of  660  yards  at  2^.  6d  1 
2s,=^£.     10)660 
6d.  =  J  of  2;?.    4)66  cost  at  2a. 
16  105. 


Ans,  £82  10^. 


201.  When  the  price  in  shillings  is  less  than  20, 

Multiply  hy  half  the  number  of  shillings,  and  the  figures  to 
the  left  of  the  right  hand  figure  will  express  the  pounds,  and 
this  figure  doubled  will  be  the  shillings. 


11.  What  is  the  cost  of  56  yards  of  cloth,  at  16^.  per  yard  ? 
16^.  =if  of  a  £:  Hence 
^^  X  ^  =  the  amount  in  pounds. 
But    56  X  if  =  56  X  3^  =  £44.8,   in    which   the    right 
hand  figure  8  expresses  tenths  of  pounds,  and  by  doubling  it, 
we  obtain  twentieths  of  pounds,  or  shillings :  therefore,  the 
reason  of  the  rule  is  manifest. 


Quest. — 201.  When  the  price  is  in  shiUings  and  less  than  20,  how  will 
you  find  tlie  cost  ?     What  is  the  rejison  of  the  rule  ? 

10 


218  '  PRACTICE. 

12.  What  will  be  the  cost  of  4514  yards  of  cloth  at  £2 
17 s.  7^d,  per  yard  ? 

4514 

. 2_ 

Cost  at  £2 9028 

4514  X  81  gives     -     -     -     3836  18^. 
at  6d.  =  J^  of  £      -     -     -       112  17;?. 

l^d.  =  1  of  6(^.  -     -     -     - 28    4;?.  3d. 

Total  cost     £13005  19;y.  3d. 


GENERAL    EXAMPLES. 

13.  What  will  19cwt,  3qr,  Ulb,  of  hops  cost,  at  £4  Us. 
9d.  per  cwt,  1 

14.  19cw;^.  3qr,  I9lb,  of  sugar,  at  £2  4s.  8d.  per  cwt.  ? 

15.  llcwt.  Iqr.  16lb.  of  soap,  at  £3  Is.  per  cwt,  ? 

16.  9cwt.  3qr.  \Olb.  of  treacle,  at  £118^.  9c?.  per  cwt.  ? 

17.  What  is  the  cost  of  40ZZ>.  of  soap,  at  Q^cts.  per  pound? 

18.  What  is  the  cost  of  70  yards  of  tape,  at  2\cts,  per 
yard? 

19.  What  is  the  cost  of  876  bushels  of  apples,  at  62 ^c^^. 
per  bushel  1 

20.  What  is  the  cost  of  1000  quills,  at  ^  cent  per  quill? 

21.  What  is  the  cost  of  1800  lead  pencils,  at  6  cents 
apiece  ? 

22.  What  is  the  cost  of  9T.  \3cwt.  \9lb,  of  pewter,  at  £14 
15^.  9d.  per  ton? 

23.  3qr.  19Ib.  lOoz.,  at  £11  12^.  d^d.  per  cwt.l 

24.  74o;^.  2pwt,  I2gr,  of  silver,  at  4^.  11-J-J.  per  oz,  ? 

25.  A  pair  of  chased  silver  salts,  weight  7oz,  llpwt.,  at 
8^.  U^d.  per  oz.1 

26.  57loz.  Upwt.  I6}gr.,  at  £3  11^.  91c?.  per  oz.  ? 

27.  What  will  be  the  cost  of  85i  yards  of  cloth,  at  $9^ 
per  yard  ? 

28.  What  will  be  the  cost  of  1848  yards  of  linen,  at  87 J 
cents  per  yard  ? 


PRACTICE.  219 

29.  What  is  the  cost  of  511  tons  of  hay,  at  #12,50  per  ton  ? 

30.  What  is  the  cost  of  693  yards  of  linen,  at  15cts.  per 
yard? 

31.  What  is  the  rent  of  725 J..  2i^.  19P.  of  land,  at  £2 
11^.  9c?.  per  acre  ? 

32.  51^1.  3E.  15P.  at  £4  10^.  per  acre  ? 

33.  97A.  14P.  at  ^£3  11^.  10c?.  per  acre? 

34.  What  is  the  cost  of  28^  yards  of  cloth,  at  $4|  per 
yard? 

35.  What  will  be  the  cost  of  2000  quills,  at  ^  cent  per 
quill  ? 

36.  What  will  154^  tons  of  hay  come  to,  at  $12  per  ton? 

37.  Wha*  is  ^he  cost  of  514yd.  3qr.  2na.,  at  17^.  9^d,  per 
vard? 

38.  125^.  E,  iqr,  Ina.  at  £1  11^.  9^  per  ell? 

39.  What  will  be  the  cost  of  1752  bushels  of  apples,  at 
62^  cents  per  bushel  ? 

40.  What  is  the  cost  of  280  yards  of  tape,  at  2^  cents  per 
yard  ? 

41.  What  is  the  cost  of  120  pounds  of  soap,  at  6^  cents 
per  pound  ? 

42.  What  cost  l7E,Fr.  Iqr,  3na,  of  Brussels  lace,  at  £3 
19s,  lli(i.  per  ell? 

43.  3i9E,Fl,  Iqr.  3na,  of  holland,  at  £1  Us.  6d.  per  ell? 

44.  475yd.  3qr,  2na.  at  £1  14^.  9\d,  per  ell  English? 

45.  375fE.^.  at  18^.  llf  J.  per  yard? 

46.  what  will  be  the  cost  of  2hhd,  5gaL  3qt,  2gi,  of  mo- 
lasses, at  12^  cents  per  quart? 

47.  What  will  be  the  cost  of  376  yards  of  cloth,  at  $1^ 
per  yard  ? 

48.  What  will  be  the  cost  of  Ihhd,  2gal,  3qt.  Ipt.  Igi.  of 
brandy,  at  56^  cents  per  quart  ? 

49.  What  will  be  the  cost  of  27ha.  3pk.  6qt.  Ipt.  of  wheat 
at  $1,75  per  bushel? 


220  TARE    AND    TRET. 


TARE  AND  TRET. 

202.  Tare  and  Tret  are  allowances  made  in  selling  goods 
by  weight. 

Draft  is  an  allowance  on  the  gross  weight  in  favor  of  the 
buyer  or  importer :  it  is  always  deducted  before  the  Tare. 

Tare  is  an  allowance  made  to  the  buyer  for  the  weight  of 
the  hogshead,  barrel  or  bag,  &:c.,  containing  the  commodity 
sold. 

Gross  Weight  is  the  whole  weight  of  the  goods,  together 
with  that  of  the  hogshead,  barrel,  bag,  &c.,  which  contains 
them. 

Siittle  is  what  remains  after  a  part  of  the  allowances  have 
been  deducted  from  the  gross  weight. 

Net  Weight  is  what  remains  after  all  the  deductions  are 
made. 

EXAMPLES. 

1.  *What  is  the  net  weight  of  25  hogsheads  of  sugar,  the 
gross  weight  being  66cwt.  3qr.  lAlb. ;  tare  lllb.  per  hogs- 
head ?. 

cwt.  qr.    lb, 
66     3     14  gross. 
25  X  11  =  275Z6.  -  -     2     1     23  tare. 


Ans. 

net. 

2.  If  the  tare  be  4lb. 
6T,2cwt.  3qr.  UIb.1 

per  hundred,  what  will  be  the 

tare  on 

Tare  for  6T.  or  I20cwt. 

=  4801b. 

2cwt. 

=      8 

3qr. 
I4lb. 

=      3 
=      0^ 

Tare    -    -    - 

-    491| 

Quest.— 202.  What  are  Tare  and  Tret  ?     What  is  Draft  ?    What  if 
Tare  ?    What  is  Gross  Weight  ?     What  is  Suttle  ?     What  is  Net  Weight  7 


TARE    AND  TRET.                                         221 

3.    What   is   the   tare   on   32    boxes  of  soap,  weighing 

315501b.,  allowing  4lb.  per  box  for  draft  and  I2lb,  in  every 
hundred  for  tare  ? 

31550  gross.  31422 

32  X  4=     128  draft.  12 

31422  3770.64 


Ans. 


4.  What  will  be  the  cost  of  3  hogsheads  of  tobacco  at 
$9,47  per  cwt.  net,  the  gross  weight  and  tare  being  of 

cwt.  qr.  lb,  lb. 

No.  1    -   -      9     3     25    -   -   tare  146 

"     2   '-   -    10     2     12    -   -      "     150 

"     3    -   -    11     1     25    -    -      "     158 

Ans.  

5.  At  £1  5^.  per  cwt.  net;  tare  4lb.  per  cwt.  :  what  will 
be  the  cost  of  4  hogsheads  of  sugar,  weighing  gross, 

cwt.  qr.    lb. 
No. 


1  -    -   -    10 

2  .    -   -    12 

3  -    -   -    13 

4  -   -   -    11 

49 
.  per  cwt      1 

3 
5 

1 
2 

0 
3 

6 
19 
10 

7 

14  gross. 
0  8oz. 

47 

1 

13  8dz.  net. 

Ans.  

6.  At  21  cents  per  lb.,  what  will  be  the  cost  of  5hhd.  of 
coffee,  the  tare  and  gross  weight  being  as  follows : 

No. 


cwt. 

qr.    lb. 

Ib. 

1    -   -     6 

2      14    -    - 

tare    94 

2    -    -     9 

1     20    -    - 

"     100 

3    -    -     6 

2     22    -    - 

"       88 

4    -   -     7 

2     25    -    - 

"       89 

5    -   -     8 

0     13    -^- 

"     100 

Ans.  

7.  At  £7  5s.  per  cwt.  net,  how  much  will  IShhd.  of  sugar 
come  to,  each  weighing  gross  8cwt.  3qr.  lib. ;  tare  12Z^.  per 
cwt. "? 


222  TARE    AND    TRET. 

8.  What  is  the  net  weight  of  I8hhd.  of  tobacco,  each 
weighing  gross  8cwt,  3qr,  I4lb. ;  tare  I6lb.  to  the  cwt.l 

9.  In  4T.  3cwt.  3qr.  gross,  tare  20lb,  to  the  cwt.,  what  is 
the  net  weight  1 

10.  What  is  the  net  weight  and  value  of  80  kegs  of  figs, 
gross  weight  7T.  llcwt.  3qr.,  tare  I4lb.  per  cwt.j  at  $2,31 
per  cwt.  ? 

11.  A  merchant  bought  19cwt.  \qr.  27lb.  gross  of  tobacco 
in  leaf,  at  $24,28  per  cwt. ;  and  12ct^^.  3qr.  I9lb.  gross 
in  rolls,  at  #28,56  per  cwt. ;  the  tare  of  the  former  was 
1491b.,  and  of  the  latter  48^lb. :  what  did  the  tobacco  cost 
him  net  ? 

12.  A  grocer  bought  I7^hhd,  of  sugar,  each  lOcwt.  Iqr, 
I4lb.,  draft  7lb.  per  cwt.,  tare  4lb.  per  104/5.  What  is  the 
value  at  $7,30  per  cwt.  net  ? 

13.  In  29  parcels,  each  weighing  3cwt.  3qr.  14lb.  gross, 
draft  Sib.  per  cwt.,  tare  4lb.  per  I04lb.,  how  much  net  weight, 
and  what  is  the  value  at  $7,50  per  cwt.  net  ? 

14.  A  merchant  bought  7  hogsheads  of  molasses,  each 
weighing  4cwt.  3qr.  I4lb.  gross,  draft  17Z5.  per  cwt.,  tare  8lb. 
per  hogshead,  and  damage  in  the  whole  99^lb.  What  is  the 
value  at  $8,45  per  cwt.  net  ? 

15.  The  net  value  of  a  hogshead  of  Barbadoes  sugar  was 
$22,50  ;  the  custom  and  fees  $12,49,  freight  #5,1 1,  factorage 
$1,31  ;  the  gross  weight  was  llcwt.  Iqr.  15lb.,  tare  ll^lb. 
per  cwt.  What  was  the  sugar  rated  at  per  cwt.  net.  in  the 
bill  of  parcels  ? 

16.  In  7hhd.  of  oil,  each  weighing  3cwt.  2qr.  14lb.  gross, 
tare  21lb.  per  cwt.,  how  many  gallons  net,  and  what  is  the 
value  at  $1,24  per  gallon? 

17.  I  have  imported  87  jars  of  Lucca  oil,  each  containing 
47  gallons  :  what  came  the  freight  to  at  $1,19  per  cwt.  net 
reckoning  lib,  in  lllb.  for  tare,  and  9lb.  of  oil  to  the  gallon  ^ 


PERCENTAGE.  223 


PERCENTAGE. 

203.  The  term  per  cent  comes  from  per  centum,  and 
means  by  the  hundred.  The  term  is  generally  used  to  ex- 
press the  interest  on  money,  but  may  also  be  employed  to 
designate  hundredth  parts  of  other  things.  Thus,  when  we 
say  twenty  per  cent  of  a  bushel  of  wheat,  we  mean  twenty 
hundredths,  or  one-fifth  of  it. 

204.  The  rate  per  cent  may  always  be  expressed  by  a 
decimal  fraction.  Thus,  five  per  cent  may  be  expressed  by 
.05,  eight  per  cent  by  .08,  fifteen  per  cent  by  .15,  &c. 

Hence,  to  find  the  amount  of  percentage  on  any  number, 
Multiply  the  number  by  the  rate  per  cent,  expressed  in  a  deci- 
mal fraction,  and  the  product  will  be  the  percentage. 

EXAMPLES. 

1.  A  has  $852  deposited  in  the  bank,  and  wishes  to  draw 
out  5  per  cent  of  it :  how  much  must  he  draw  for  ? 

2.  A  merchant  has  1200  barrels  of  flour ;  he  shipped  64  per 
cent  of  it  and  sold  the  remainder  :  how  much'  did  he  sell  ? 

3.  A  merchant  bought  1200  hogsheads  of  molasses.  On 
getting  it  into  his  store,  he  found  it  short  3J  per  cent :  how- 
many  hogsheads  were  wanting  1 

4.  Two  men  had  each  $240.  One  of  them  spends  14  per 
cent,  and  the  other  18^  per  cent:  how  many  dollars  more 
did  one  spend  than  the  other  ? 

5.  What  is  the  difference  between  5J  per  cent  of  $800 
and  6i  per  cent  of  $1050  ? 

6.  A  trader  laid  out  $160  as  follows  :  he  pays  24  per  ct.  of 
his  money  for  broadcloths ;  30  per  ct.  of  what  is  left  for  linens ; 
12  per  ct.  of  what  is  left  for  calicoes ;  and  then  5  per  ct.  of 
the  residue  for  cottons :  how  much  did  he  pay  for  cottons  ? 

Quest. — 203.  What  do  you  understand  by  the  term  per  cent  ?  For  what 
IS  the  term  generally  used  ?  What  do  you  understand  by  twenty  per  cent? 
What  by  eight  per  cent?  204.  How  may  the  rate  per  cent  be  expressed? 
How  do  you  express  five  per  cent  ?  Eight  per  cent  ?  How  do  you  find 
the  amount  of  percentage  on  any  given  number? 


224  PERCENTAGE. 

7.  A  man  purchased  250  boxes  of  oranges,  and  found  that 
ne  had  lost  in  bad  ones  18  per  cent :  to  how  many  full  boxes 
were  his  good  oranges  equal  ? 

8.  If  I  buy  895  gallons  of  molasses  and  lose  17  per  cent 
by  leakage,  how  much  have  I  left  ? 

205.  To  find  the  per  cent  which  one  number  is  of  another. 

If  I  buy  6  hogsheads  of  molasses  for  $200  and  sell  them 
f  >r  $220,  what  do  I  gain  per  cent,  on  the  money  expended? 

It  is  plain  that  $20  is  the  amount  made.     What  per  cent 
U  $20  of  $200  ;  that  is,  how  many  hundredths  of  $200  1     If 
we  add  two  ciphers  to  the  first,  and  then  divide  it  by  the 
Becond,  the  quotient  will  express  the  hundredths*.     Thus, 
2000 

that  is,  20  is  ten  per  cent  of  200. 

Hence,  to  determine  the  per  cent  which  one  number  is  of 
another, 

I.  Bring  the  number  which  determines  the  per  cent  to  hun* 
dredths  by  annexing  two  ciphers  or  removiiig  the  decimal  point 
two  places  to  the  right. 

II.  Divide  the  number  so  formed  by  the  number  on  which  the 
percentage  is  estimated^  and  the  quotient  will  express  the  per  cent* 

EXAMPLES. 

1.  A  man  has  $550  and  purchases  goods  to  the  amount 
of  $82,75 :  what  per  cent  of  his  money  does  he  expend  ? 

2.  A  merchant  goes  to  New  York  with  $1500 ;  he  first  lays 
out  20  per  ct.,  after  which  he  expends  $660 :  what  per  ct.  was 
his  last  purchase  of  the  money  that  remained  after  his  first  ? 

3.  Out  of  a  cask  containing  300  gallons,  60  gallons  are 
drawn  :  what  per  cent  is  this  1 

4.  If  I  pay  $698,33  for  3  hhds.  of  molasses  and  sell  them  for 
$837,996,  how  much  do  I  gain  per  ct.  on  the  money  laid  out  ? 

5.  If  I  pay  $698,33  for  3  hhds.  of  sugar  and  sell  them  for 
$837,996,  how  much  do  I  make  per  ct.  on  the  amount  received  ? 

Quest. — ^205.  How  do  you  find  the  per  ct.  which  one  number  is  of  another  1 


SIMPLE    INTEREST.     .  225 


SIMPLE  INTEREST. 

206.  Interest  is  an  allowance  made  for  the  use  of  money 
that  is  borrowed. 

For  example,  if  I  borrow  $100  of  Mr.  Wilson  for  one 
year,  and  agree  to  pay  him  $6  for  the  use  of  it,  the  $6  is 
called  the  interest  of  $100  for  one  year,  and  at  the  end  of 
the  time  Mr.  Wilson  should  receive  back  his  $100  together 
with  the  $6  interest,  making  the  sum  of  $106. 

The  money  on  whi^h  interest  is  paid,  is  called  the  Prin- 
cipal, 

The  money  paid  for  the  use  of  the  principal,  is  called  the 
Interest. 

The  principal  and  interest,  taken  together,  are  called  the 
Amount. 

In  the  abave  example, 

$100     is  the  principal,' 
$     6     is  the  interest,  and 
$106     is  the  amount. 

The  interest  of  $100  for  one  year,  determines  the  rate  of 
interest,  or  rate  per  cent.  In  the  example  above,  the  rate  of 
interest  is  6  per  cent,  or  $6  for  the  use  of  the  hundred.  Had 
$8  been  paid  for  the  use  of  the  $100,  the  rate  of  interest 
would  have  been  8  per  cent ;  or  had  $3  only  been  paid,  the 
rate  of  interest  would  have  been  3  per  cent. 

Legal  interest  is  the  rate  of  interest  established  by  law.  In 
the  New  England  States,  and  indeed  in  most  of  the  other 
states,  the  legal  interest  is  6  per  cent  per  annum,  that  is,  6 
per  cent  by  the  year. 

Quest. — 206.  What  is  Interest?  What  is  the  money  called  on  which 
interest  is  paid  ?  Wiiat  is  the  money  called  which  is  paid  for  the  use  of  the 
principal?  What  is  the  amount?  What  determines  the  rate  of  interest? 
What  is  legal  interest  ?  What  is  meant  by  per  annum  ?  How  much  is  the 
interest  per  annum  in  most  of  the  states?  What  is  it  in  New  York?  In 
Alabama? 

f  I   "«J?iaS  Jfe(9p  s«  xpAV  St?  s.T9qqoj 


p  JO  aiu^tlifo  9T{;  pjB^aj  ^tjqAvaoios 
pui  )i    *ifa^nnoo  eqi  to  uonjod  poii 


0% 
9A 


226  SIMPLE    INTEREST. 

In  New  York,  however,  it  is  7,  and  in    Alabama    8  per 
cent. 


207.  To  find  the  interest  on  any  given  principal  for  one  or 
more  years. 

The  interest  of  each  dollar,  for  a  single  year,  will  be  so 
many  hundredths  of  itself  as  are  expressed  by  the  rate  of  in- 
terest. Thus,  if  the  rate  of  interest  be  4  per  cent,  each  dol- 
lar will  produce  annually  an  interest  of  .04  of  a  dollar,  or  4 
cents :  if  the  rate  be  5  per  cent,  it  will  produce  .05  of  a  dol- 
lar, or  5  cents :  if  6  per  cent,  .06,  or  6  cents,  &c. 

Hence,  to  find  the  interest  on  any  given  sum  for  one  or 
more  years. 

Multiply  the  principal  hy  the  decimal  fraction  which  expresses 
the  rate  of  interest,  and  the  product  so  arising  hy  the  number 
of  years.     Or, 

Multiply  the  decimal  fraction  which  expresses  the  rate  of  in- 
terest hy  the  number  of  years,  and  then  multiply  the  principal 
hy  this  product. 

EXAMPLES. 

1.  What  is  the  interest  on  $1960  for  four  years,  at  7  per 
cent  per  annum  1 


The  rate  of  interest  being 
7  per  cent,  each  dollar  will 
produce  .07  of  a  dollar,  or  7 
cents,  in  one  year:  hence, 
$137,20  will  be  the  interest 
on  the  sum  for  one  year,  and 
$548,80  for  4  years. 


OPERATION. 

$1960 
.07 


$137,20  int.  for  1  year. 

4  number  of  years. 
$548,80  Ans, 


Quest. — 207.  What  will  be  the  interest  of  one  dollar  for  one  year  ?  What 
will  express  decimally  the  interest  on  one  dollar  for  one  year  at  4  per  cent  ? 
What  will  express  it  at  5  per  cent  ?  At  6  ?  At  7  ?  At  8  ?  How  do  you 
find  the  interest  on  any  sum  for  one  or  more  years  ?  What  will  be  the 
multiplier  when  the  rate  of  interest  is  4  per  cent,  and  the  time  3  years  ? 
When  the  rate  is  6  per  cent  and  the  time  5  years  ?  When  the  rate  is  8 
per  cent  and  the  time  3  years? 


SIMPLE    INTEREST.  227 


OPERATION. 

78,457 
.05  X  3  =         .15 

392285 
78457 
Ans.   #11,76855 


2.  What  is  the  interest  on  $78,457  dollars  for  three  years, 
at  5  per  cent  per  annum  ? 

Since  there  are  three  places 
of  decimals  in  the  multipli- 
cand arid  two  in  the  multi- 
plier, there  will  be  five  in  the 
product  (Art.  149).  Observe 
that  the  two  first,  counting  from 
the  comma  to  the  right,  are 
cents,  the  third  mills,  the  fourth  tenths  of  mills,  &c. 

3.  What  is  the  interest  on  $365,874  for  one  year,  at  5^ 
per  cent  ? 

We  first  find  the  interest  at 
5  per  cent,  and  then  the  in- 
terest for  i  per  cent :  the 
sum  is  the  interest  at  5J  per 
cent. 


OPERATION. 

$365,874 
.05 


18,29370 
1,82937  iper  cent. 


$20,12307  Ans. 

4.  What  is  the  interest  on  $2871,24  for  6  years,  at  7  pel 
cent? 

5.  What  is  the  interest  on  $535,50  for  25  years,  at  6  per 
cent? 

^      6.  What  is  the  interest  on  $1125,819  for  5  years,  at  8  per 
cent  per  annum  ? 

7.  What  is  the  interest  on  $8089,74  for  12  years,  at  5 
per  cent  ? 

8.  What  is  the  interest  on  $1226,35  for  7  years,  at  7^  per 
cent? 

9.  What  is  the  interest  on  $3153,82  for  9  years,  at  4^  per 
cent? 

10.  What  is  the  interest  on  $982,35  for  4  years,  at  6  per 
cent? 

11.  What  is  the  interest  on  $1914,16  for  18  years,  at  3| 
per  cent? 

12.  What  is  the  interest  on  $2866,28  for  6  years,  at  8  per 
cent? 


228 


SIMPLE    INTEREST. 


13.  What  is  the  interest  on  $16199,48  for  16  years,  at  5 
per  cent  ? 

14.  What  is  the  interest  on  $897,50  for  21  years,  at  6 
per  cent  ? 

CASE    II. 

208.  To  find  the  interest  for  any  number  of  months,  at  the 
rate  of  6  per  cent  per  annum. 

At  the  rate  of  6  per  cent  per  annum,  one  month  produces 
J  per  cent  on  the  principal ;  and  hence,  every  two  months 
produces  one  per  cent  on  the  principal.  Therefore  to  find 
the  interest  for  months. 

Divide  the  number  of  months  by  2  and  regard  the  quotient  as 
hundredths.  Then  multiply  the  principal  by  the  decimal  so 
found,  and  the  product  will  be  the  interest. 


EXAMPLES. 

1.  What  is  the  interest  on  $651  for  8  months,  at  6  per 
cent  per  annum  ? 

The    decimal   corre-  operation. 

sponding  to   8  months,  $651 

which  gives  4  per  cent,  .04       half  the  number  of  months, 

is  .04:  hence,  the   in-         $26,04 
terest  is  $26,04. 

2.  What  is  the  interest  on  $614,364  for  9  months,  at  6  per   * 
cent  per  annum  ? 


The  decimal  corresponding  to  9 
months  is  .04^,  and  hence  the  in- 
terest is  $27,64638. 


OPERATION. 

$614,364 

.04| 
2457456 
307182 
$27,64638 


Quest. — ^208.  At  the  rate  of  6  per  cent,  what  will  be  the  interest  on  any 
principal  for  one  month  ?  What  time  will  produce  one  per  cent  ?  How  do 
you  find  the  interest  on  any  principal  for  any  number  of  months  ?  What 
is  the  multiplier  for  4  months?  What  for  6  months ?  What  for  7  ?  What 
for  8  ?     For  9  ?    What  for  10  ?     For  11  ?    What  for  12  ? 


SIMPLE    INTEREST.  229 

3.  What  is  the  interest  on  $17507,30  for  14  months,  at  6 
per  cent  ? 

4.  What  is  the  interest  on  $982,41  for  9  months,  at  6  per 
cent  ? 

5.  What  is  the  interest  on  $75192,84  for  16  months,  at  6 
per  cent  ? 

6.  What  is  the  interest  on  $7953,70  for  9  months,  at  6  per 
cent? 

7.  What  is  the  interest  on  $15907,40  for  27  months,  at  6 
per  cent? 

8.  What  is  the  interest  on  $4918,50  for  11  months,  at  6 
per  cent? 

9.  What  is  the*  interest  'on  $84377,91  for  7  months,  at  6 
per  cent  ? 

10.  What  is  the  interest  on  $91358,24  for  17' months,  at 
6  per  cent  ? 

11.  What  is  the  interest  on  $31573,25  for  10  months,  at  6 
per  cent? 

12.  What  is  the  interest  on  $959875,45  for  18  months,  at 
6  per  cent  ? 

CAS5  ni. 

209.  To  find  the  interest  at  6  per  cent  per  annum,  for  any 
number  of  days. 

In  computing  interest  the  month  is  reckoned  at  30  days. 
Hence,  60  days,  which  make  two  months,  will  give  an  in- 
terest of  one  per  cent  on  the  principal,  and  consequently,  6 
days  will  give  an  interest  of  one  mill  on  the  dollar,  or  one- 
thousandth  of  the  principal.  If,  therefore,  the  days  be  divided 
by  6,  the  quotient  will  show  how  many  thousandths  of  the 
principal  must  be  taken  on  account  of  the  days.  Hence,  to 
find  the  interest  for  any  number  of  days  less  than  60, 

Quest. — 209.  In  computing  interest  for  days,  at  what  is  the  month 
reckoned  ?  How  many  days  give  one  per  cent  ?  What  part  of  the  prin- 
cipal is  one  per  cent?  How  many  days  will  give  one-thousandth  of  the 
principal  ?  How  will  you  find  how  many  thousandths  of  the  principal  must 
be  taken  for  the  days  ? 


230 


SIMPLE    INTEREST. 


Divide  the  days  hy  6,  and  multiply  the  principal  hy  the  quO' 
tientf  considered  as  thousandths. 


EXAMPLES. 

1.  What  is  the  interest  on  $297,047  for  28  days,  at  6  per 
cent  per  annum  ?  operation. 

We  find  that  the  28  days  give  $297,047 

4|-  thousandths.  We  muUiply  the 
principal  by  .004,  and  then  add  |- 
of  the  principal  multiplied  by  one- 
thousandth  for  the  fractional  part. 


28^6=4f. 


Addl 

3 


004f 

1188188 

99015 

99015 

$1,386218 


210.  To  avoid  the  fractions  which  sometimes  appear  in 
the  multipliers,  we  may,  if  we  please,  first  multiply  the  prin- 
cipal by  the  number  of  days,  and  then  divide  the  product  by 
6,  which  will  give  the  same  quotient  as  found  above.  Hence, 
to  find  the  interest  for  any  number  of  days. 

Multiply  the  principal  by  the  number  of  days,  divide  the 
product  by  6,  and  then  point  off  in  the  quotient  three  more  places 
for  decimals  than  there  are  decimals  in  the  given  principal. 

2.  What  is  the  interest  on  $657,87  for  13  days,  at  6  per 
cent  per  annum  ? 

We  first  multiply  the  given  prin- 
cipal by  13  ;  we  then  divide  the 
product  by  6  ;  and  since  there  are 
two  places  of  decimals  in  the  prin- 
cipal, we  point  off  five  in  the  quo- 
tient. 


OPERATION. 

$657,87 
13 


197361, 
65787 
6)855231 
$1,42538 


Note — Let  each  of  the  following  examples  be  worked  by  both 
methods ;  though,  when  the  days  exceed  60,  the  second  method  is 
preferable. 

Quest. — How  do  you  find  the  interest  for  less  than  60  days  ?  What  is  the 
maltipUer  for  6  days  ?  For  9  days  ?  For  10  days  ?  For  15  days  ?  For  20 
days?  For  25  days?  210.  How  may  the  interest  for  days  be  found  by 
the  second  method? 


SIMPLE    INTEREST.  231 

3.  Find  the  interest  on  $785,469  for  25  days.  Also,  the 
interest  on  $8709,27  for  100  days. 

4.  What  is  the  interest  on  $2691,12  for  150  days^ 

5.  What  is  the  interest  on  $1151,44  for  29  days  ? 

6.  What  is  the  interest  on  $136,25  for  19  days? 

7.  What  is  the  interest  on  $981,90  for  70  days  ? 

8.  What  is  the  interest  on  $757,06  for  9  days  ? 

9.  What  is  the  interest  on  $864  for  95  days  ? 

10.  What  is  the  interest  on  $11268,75  for  17  days? 

11.  What  is  the  interest  on  $4428,10  for  165  days? 

12.  What  is  the  interest  on  $975,95  for  14  days  ? 

13.  What  is  the  interest  on  $28793,28  for  127  days  ? 

21 1 .  Note. — The  above  method  of  computing  interest  for  days, 
is  the  one  in  general  use.  It,  however,  considers  the  year  as  made 
up  of  360  instead  of  365  days ;  and  hence  the  result  is  too  large  by 
5  of  the  365  parts  into  which  the  interest  found  may  be  divided. 
Hence,  the  interest  found  will  be  too  large  by  its  ^^3  part,  by  which 
it  must  be  diminished  when  entire  accuracy  is  desired. 

CASE    IV. 

212.  To  find  the  interest  at  6  per  cent  per  annum  for  years, 
months,  ajid  days. 

Find  the  interest  for  the  years  by  Case  I-,  for  the  months 
by  Case  II.,  and  for  the  days  by  Case  III, ;  then  add  the 
several  results  together^  and  their  sum  will  be  the  answer 
sought.     Or, 

Form  a  single  multiplier  for  the  years,  months,  and  days, 
and  then  multiply  the  principal  by  it. 

EXAMPLES. 

1.  What  is  the  interest  on  $1597,27  at  6  per  cent,  for  3 
years  9  months  and  1 1  days  ? 

Quest. — 21 1 .  How  many  days  does  the  above  method  give  to  the  year  ? 
Is  the  result  obtained  too  great  or  too  small  ?  By  how  much  is  it  too 
great?  How  will  you  find  the  exact  interest?  212.  How  do  you  find  the 
interest  at  6  per  cent  per  annum  for  years,  months,  and  days  ?  What  is 
the  multiplier  for  1  year  4  months  and  12  days  ?  What  for  2  years  8 
months  and  18  days  ?     For  3  years  10  months  and  24  days  ? 


232 


SIMPLE    INTEREST. 


1st  methoi 

). 

$1597,27 

$1597,27 

$1,59727  for    6  days, 

06X3==         ,18 

.04^ 

,79863  for    3  days. 

1277816 

638908 

,53242  for    2  days. 

159727 

798631. 

$2,92832  for  11  days, 

$287,5086 

$71,8771i 

Interest  for    3  years 

$287,5086 

li 

"      9  months 

71,8771  + 

a 

"    11  days 
Total  interest 

2,9283  + 
$362,3140+ 

2d    METHOD. 

Multiplier  for    3  years  =:  .06  X  3  =  .18. 
"  "      9  months  =  .045. 

"  "11  days=r  Li  =  .OOlf. 


Entire  multiplier         0.226|-. 
Then,     $1597,27  X  0.226|-  =z  $362,3140  +  . 

2.  What  is  the  interest  on  $252803,87  for  1  year  1  month 
ind  1  day  ? 

3.  What  is  the  interest  on  $3195,54  for  7  years *6  months 
and  22  days  ? 

4.  What  is  the  interest  on  $1352,25  for  4  years  and  7 
months  ? 

5.  What  is   the   interest  on  $23518,20  for   9   years,   11 
months,  and  16  days? 

6.  What  is  the  interest  on  $2420,70  for  1  year  and  10 
months  ? 

7.  What  is  the  interest  on  $19574  for  12  years  and  1  day  ? 

8.  What  will  be  the  amount  of  $1947,66  after  21   years 
and  8  months  ? 

9.  What  is  the  interest  on  $1330,50  for  14  years,  4  months, 
dnd  24  days  ? 

10.  What  is  the  interest  on  $3227,60  for  2  years,  8  months 
and  20  days  ? 


SIMPLE    INTEREST.  233 

11.  What  is  the  interest  on  $79265,375  for  8  years  7 
months  and  6  days  ? 

12.  What  will  be  the  amount  of  $9537,15  after  11  years, 
2  months,  and  18  days  ? 

CASE    V. 

213.  To  find  the  interest  when  there  are  months  and  days, 
and  the  rate  of  interest  is  greater  or  less  than  6  per  cent. 

Find  the  interest  at  6  per  cent.  Then  add  to,  or  subtract 
from  the  interest  so  found ^  such  part  of  it,  as  the  given  rate 
exceeds  or  falls  short  of  6  per  cent. 

EXAMPLES. 

1.  What  is  the  interest  on  $179,25,  at  7  per  cent  per  an- 
num, for  S  years  and  4  months  1 

Multiplier  for  3  years  =  .06  X  3  .=  .18 

"  "    4  months  =  .02 

Entire  multiplier         .20 

Hence,  $179,25  x  .20=  $35,8500  interest  at  6  per  cent. 
Add  i  5,9750 

$4]  ,8250  interest  at  7  per  cent. 


2.  What  is  the  interest  on  $974,50  for  9  years,  6  months, 
and  18  days,  at  4  per  cent  per  annum? 

Multiplier  for    9  years  at  6  per  cent  =  9  X  .06  =  .54 
"  "      6  months  =  .03 

"  "    18  days  =  18 -^  6  =3  =.003 

Entire  multiplier        -       -       -       -         .573 

Hence,  $974,50  x  .573  =  $558,3885 
Subtract  one-third  186,1295 
Int.  at  4  per  cent     $372,2590 


3.  What  is  the  interest  on  $874,42,  at  3  per  cent,  for  19 
years  and  6  months  1 

Quest. — ^213.  How  do  you  find  the  interest  when  there  are  months  and 
days,  and  the  rate  greater  £han  6  per  cent?  How  do  you  find  the  in- 
terest when  it  is  less  ? 


234  SIMPLE    INTEREST. 

4.  What  is  the  interest  on  $358,50,  at  7  per  cent,  for  6 
years  and  8  months  ? 

5.  What  is  the  interest  on  $1975,98,  at  5  per  cent,  for  10 
years  4  months  and  1 8  days  ? 

6.  What  is  the  interest  on  $1461,75  for  4  years  and  9 
months,  at  8  per  cent  I, 

7.  What  is  the  interest  on  $45000  for  1  year  and  4 
months,  at  7  per  cent  ? 

8.  What  will  be  the  total  amount  of  $2238,96  after  2 
years  and  7  months,  at  7  per  cent  ? 

9.  What  is  the  interest  of  $1200  for  1  month  and  12  days, 
at  5  per  cent  ? 

10.  What  is  the  interest  on  $1064,82  for  6  years  and  6 
months,  at  4^  per  cent  ? 

11.  What  is  the  interest  on  $1752,96,  at  7  per  cent,  for  4 
years  9  months  and  14  days  ? 

12.  What  is  the  interest  on  $17518,54,  at  7^  per  cent,  for 
3  years  and  9  days  ? 

13.  What  is  the  interest  on  $15138,22  for  3  years,  4 
months  and  18  days  at  6^  per  cent  per  annum  ? 

14.  What  is  the  interest  on  $4l87,635,  at  5  per  cent,  for 
5  years  5  months  and  5  days  1 

15.  What  is  the  interest  on  $167,50  for  7  months  and  17 
days,  at  7  per  cent  per  annum  ? 

16.  What  is  the  interest  on  $2934,25  for  2  years  8  months 
and  19  days,  at  8^  per  cent? 

17.  What  is  the  interest  on  $19345,31,  at  4^  per  cent,  for 
5  years  6  months  and  15  days  ? 

214.  Note. — In  computing  interest,  it  is  often  very  convenient 
to  find  the  interest  for  the  months  by  considering  them  as  aliquot 
parts  of  a  year,  and  the  interest  for  days  by  considering  them  as 
aliquot  parts  of  a  month. 

Quest. — ^214.  Explain  the  second  method  of  computing  interest  for 
months  and  days.  What  part  of  a  year  are  3  months  ?  Four  months  ?  Six  ? 
Eight  ?  Nine  ?  What  part  of  a  month  are  6  days  ?  Five  days  ?  Tep. 
days? 


SIMPLE    INTEREST.  235 

EXAMPLES. 

1.  What  is  the  interest  of  $806,90  for  1  year  10  months 


and  10  days,  at  6  per  cent? 

$806,90 

.06 

6)$48,4140  =z  int.  for  1  year. 

$8,069 

2)8,069     =  int.  for  2  months. 

5 

3)4,034+=  int.  for  1  month. 

$40,345  int.  1 

1,344+ =1  int.  for  10  days. 

Interest  for  1  year     -     - 

■     $48,4140 

"         "10  months    - 

40,345 

"         "    10  days  -     - 

1,344  + 

Total  interest 

$90,103  + 

2.  What  is  the  interest  of  $200  for  10  years  3  months  and 
6  days,  at  7  per  cent  ? 
$200 
.07 


4)14,00     =  int. 

for  1  year. 

$14,00 

3)3,50     =  int. 

for  3  months. 

10 

5)1,16+=  int. 

for  1  month. 

$140,00 

,23  +  =  int.  for  6  days. 

$140,00      interest  for  10  years. 
3,50      interest  for  3  months. 
,23+  interest  for  6  days. 
Ans.  $143,73  + 

3.  What  is  the  interest  of  $264,52  for  2  years  8  months 
and  20  days,  at  6  per  cent  per  annum  ? 

4.  What  is  the  interest  of  $76,50  for  1  year  9  months  and 
12  days,  at  6  per  cent  ? 

5.  What  is  the  interest  of  $1041,75  for  1  year  1  month 
and  6  days,  at  4  per  cent  per  annum  ?  Also,  at  5  per  cent  ? 
At  5i  per  cent  ?  At  6  per  cent  1  At  7  per  cent  ?  At  7|  per 
cent  ?    At  8  per  cent  ?    At  8i  per  cent  ?    And  at  9  per  cent  ? 


236 


SIMPLE    INTEREST. 


6.  What  is  the   interest,  at  6  per  cent  per  annum,  on 
$241,60,  for  3  years  3  months  and  15  days? 

7.  What  is  the  interest,   at   8  per  cent  per  annum,  on 
1351,74,  for  3  years  6  months  and  6  days  ? 

8.  What  is  the  interest,  at  7  per  cent,  on  $1761,75,  for 
5  years  5  months  and  5  days  ? 

9.  What  is   the   interest  on  $135178,40  for  3   years   9 
months  and  12  days,  at  5  per  cent  per  annum  ? 


215,  To  find  the  interest,  when  the  sum  on  which  the  in- 
terest is  to  be  cast  is  in  pounds,  shillings,  and  pence. 

I.  Reduce  the  shillings  and  pence  to  the  decimal  of  a  pound 
(Art.  161).- 

II.  Then  find  the  interest  as  though  the  sum  were  dollars  and 
cents  ;  after  which  reduce  the  decimal  part  of  the  answer  tr 
shillings  and  pence  (Art.  162). 


EXAMPLES. 


1.  What  is  the  interest,  at  6  per  cent,  of  £27  15^.  9c?.  for 


2  years  1 


We  first  find  the  interest 
for  one  year.  We  then  mul- 
tiply by  2,  which  gives  the 
interest  for  two  years.  We 
then  reduce  to  pounds,  shil- 
.  lings,  and  pence. 


OPERATION. 

£27  15^.  9d.  =  £27,7875 
.06 


1.667250 

2_ 

£3.334500 

20 

6.690000 

12 

8.280000 

4 

1.120000 
Ans,  £3  es,  S^d 


Quest — 215.  How  do  you  determine  the  interest  when  the  sum  is  in 
pounds,  shilling^,  and  pence  ? 


SIMPLE    INTEREST.  237 

2.  What  is  the  interest  on  £203.18^.  Sd.,  at  6  per  cent, 
h)r  3  years  8  months  16  days  ? 

3.  What  is  the  interest  on  £255  10^.  Sd,  at  6  per  cent,  for 
3  years  and  3  months  ? 

4.  What  is  the  interest  of  £215  13^.  Sd.,  at  6  per  cent,  for 
3  years  6  months  and  6  days  ? 

5.  What  will  £559  7^.  4c?.  amount  to  in  3  years  and  a 
half,  at  5i  per  cent  per  annum  1 

6.  What  is  the  interest  of  £1543  10^.  6d.  for  3  years  and 
a  half,  at  4  per  cent  1 

7.  What  is  the  interest  of  £1047  3^.  for  3  years  and  a 
half,  at  6  per  cent  ? 

8.  What  is  the  interest  on  £511  1^.  4c?.,  at  6  per  cent  per 
annum,  for  6yr.  6mo,  ? 

9.  What  is  the  interest  on  £161  15^.  3d,,  at  6  per  cent, 
for  7yr.  ISda.l 

APPLICATIONS. 

216.  For  computing  the  interest  on  notes,  the  time  may  be 
found  by  the  table  in  Art.  38. 

The  day  on  which  a  note  is  dated  and  the  day  on  which 
it  falls  due,  are  not  both  reckoned  in  determining  the  time , 
but  one  of  them  is  always  excluded. 

Thus,  a  note  dated*  on  the  1st  of  May,  and  falling  due 
on  the  16th  of  June,  will  bear  interest  but  one  month  and  15 
days. 

Calculate  the  interest  on  the  following  notes. 


$382,50  Philadelphia,  January  1st,  1846. 

1.  For  value  received  I  promise  to  pay  on  the  10th  day 
of  June  next,  to  C.  Hanford  or  order,  the  sum  of  three  hun- 
dred and  eighty-two  dollars  and  fifty  cents  with  interest  from 
j  date,  at  7  per  cent.  John  Liberal. 

Quest. — 216.  How  may  the  time  be  found  for  computing  interest  on 
notes?  What  days  named  in  a  note  are  reckoned  and  what  excluded,  in 
reckoning  the  time  ?  If  a  note  is  dated  on  the  first  and  payable  on  the 
15th,  how  many  days  will  the  interest  run?  • 


238  SIMPLE    INTEREST. 

$612  .  Baltimore,  January  1st,  1833. 

2.  For  value  received  I  promise  to  pay  on  the  4th  of  July, 
1835,  to  Wm.  Johnson  or  order,  six  hundred  and  twelve  dol- 
lars with  interest  at  6  per  cent  from  the  1st  of  March,  1833. 

John  Liberal. 


#3120  Charleston,  July  3d,  1846. 

3.  Six  months  after  date,  I  promise  to  pay  to  C.  Jones  or 
order,  three  thousand  one  hundred  and  twenty  dollars  with 
interest  from  the  1st  of  January  last,  at  7  per  cent 

Joseph  Springs. 


$786,50  Mobile,  July  7th,  1845. 

4.  Twelve  months  after  date,  I  promise  to  pay  to  Smith 
&  Baker  or  order,  seven  hundred  ancj  eighty-six  ^^  dollars 
for  value  receivea  with  interest  from  December  3d,  1845,  at 
8  per  cent.  Silas  Day. 


$4560,72  Cincinnati,  March  10th,  1846. 

5.  Nine  months  after  date,  for  value  received,  I  promise 
to  pay  to  Redfield,  Wright,  &  Co.  or  order,  four  thousand 
five  hundred  and  sixty  ^^  dollars  with  interest  after  6 
months,  at  7  per  cent.  Frederick  Stillman. 


$1854,83  Boston,  July  17th,  1846. 

6.  Eleven  months  after  date,  for  value  received,  we  prom- 
ise to  pay  to  the  order  of  Fondy,  Burnap,  &  Co.,  one  thou- 
sand eight  hundred  and  fifty-four  -f^-^  dollars  with  interest 
from  May  13th,  1846,  at  6  per  cent.  Palmer  <Sf  Blake, 

PARTIAL    PAYMENTS. 

217.  We  shall  now  give  the  rule  established  in  New  York 
(See  Johnson's  Chancery  Reports,  Vol.  I.  page  17)  for  com- 
puting the  interest  on  a  bond  or  note,  when  partial  payments 
nave  been  made.  Tire  same  rule  is  also  adopted  in  Massa- 
chusetts, anj  in  most  of  the  other  states. 


SIMPLE    INTEREST.  239 

# 

I.  Compute  the  interest  on  the  principal  to  the  time  of  the 
first  payment,  and  if  the  payment  exceed  this  interest,  add  the 
interest  to  the  principal  and  from  the  sum  subtract  the  payment : 
the  remainder  forms  a  new  principal. 

II.  But  if  the  payment  is  less  than  the  interest,  take  no  no- 
tice of  it  until  other  payments  are  made,  which  in  all,  shall 
exceed  the  interest  computed  to  the  time  of  the  last  payment : 
then  add  the  interest,  so  computed,  to  the  principal,  and  from 
the  sum  subtract  the  sum  of  the  payments :  the  remainder  will 
form  a  new  principal  on  which  interest  is  to  be  computed  as 
before. 

EXAMPLES. 


$349,998  Richmond,  Va.,  May  1st,  1826. 

1.  For  value  received  I  promise  to  pay  James  Wilson  or 
order,  three  hundred  and  forty-nine  dollars  ninety-nine  cents 
and  eight  mills  with  interest,  at  6  per  cent. 

James  Pay  well. 

On  this  note  were  endorsed  the  following  payments  : 
Dec.  25th,  1826  received  $49,998 
July    10th,  1827         "         $  4,998 
Sept.  1st,     1828         "         $16,008 
June   14th,  1829         "         $99,999 
What  was  due  April  15th,  1830? 
Principal  on  int.  from  May  1st,  1826        -      -      $349,998 
Interest  to  Dec.  25th,  1826,  time  of  first  pay- 
ment, 7  months  24  days 13,649  + 

^  Amount      -      -      $363,647+ 

Payment  Dec.  25th,  exceeding  interest  then 

due $  49,998 

Remainder  for  a  new  principal    -      -      -      -  $313,649 
Interest  of  $313,649  from  Dec.  25th,  1826,  to 

June  14th,  1829,  2  years  5  months  20  days  $  46,524+ 

Amount      -      -  $360,173 

Quest. — ^217.  What  is  the  rule  in  regard  to  partial  payments? 


240  SIMPLE    INTEREST.  ^ 

Payment,  July    lOth,    1827,  less)         ^  ^^^ 

than  interest  then  due     -      -      ;  ' 

Payment,  Sept.  1st,  1828  -      -      -       15,008 
Their  sum        -----      ^ 

less  than  interest  then  due    -      ^  ' 

Payment,  June  14th,  1829        -      -       99,999 
Their  sum  exceeds  the  interest  then  due        -      $120,005 
Remainder  for  a  new  principal,  June   14th, 

1829 240,168 

Interest  of  $240,168  from  June  14th,  1829,  to 

April  15th,  1830,  10  months  1  day       -      -      $  12,048 
Total  due,  April  15th,  1830   -      -      $252,216  + 


$6478,84  New  Haven,  Feb.  6th,  1825. 

2.  For  value  received  I  promise  to  pay  William  Jenks  or 
order,  six  thousand  four  hundred  and  seventy-eight  dollars 
and  eighty-four  cents  with  interest  from  date,  at  6  per  cent. 

John  Stewart, 

On  this  note  were  endorsed  the  following  payments  • 
May  16th,  1828,  received  $545,76 
May  I6th,  1830,         "         $1276 
Feb.  1st,     1831,         "         $2074,72. 

What  remained  due  August  11th,  1832  ? 

3.  A's  note  of  $7851,04  was  dated  Sept.  5th,  1837,  on 
which  were  endorsed  the  following  payments,  viz  : — Nov. 
13th,  1839,  $416,98  ;  May  10th,  1840,  $152  :  what  was  due 
March  1st,  1841,  the  interest  being  6  per  cent?* 


$8974,56  New  York,  Jan.  3d,  1842. 

4.  For  value  received  I  promise  to  pay  to  James  Knowles 
or  order,  eight  thousand  nine  hundred  and  seventy-four  dol- 
lars and  fifty-six  cents,  with  interest  from  date  at  the  rate  of 
7  per  cent.  Stephen  Jones, 

On  this  note  were  endorsed  the  following  payments : 


SIMPLE    INTEREST.  241 

Feb.  leth,  1843,  received  $1875,40 
Sept.  15th,  1844,         "         $3841,26 
Nov.  11th,  1845,         "         $1809,10 
June     9th,  1846,         "         $2421,04. 
What  was  due  July  1st,  1846  ? 

QUESTIONS    IN    INTEREST. 

218.  In  all  the  questions  relating  to  interest  four  things 
have  been  considered,  viz.: 

1st.  The  principal;  2d.  The  rate  of  interest;  3d.  The 
time ;  and  4th.  The  amount  of  interest.  Now,  these  four 
quantities  are  so  connected  with  each  other,  that  if  three  of 
them  be  known  the  fourth  can  always  be  found. 

CASE    I. 

219.  The  principal,  the  rate  of  interest,  and  the  time  being 
known,  to  find  the  interest. 

This  case  has  already  been  considered. 

CASE    II. 

220.  Having  given  the  interest,  the  time,  and  the  rate  ot 
•interest,  to  find  the  principal. 

When  the  time  and  rate  are  the  same,  the  interest  on  any 
principal,  is  to  any  other  interest,  as  the  first  principal,  is  to  the 
second ;  that  is, 

Interest  of  $1  :  given  interest  :  :  $1  :  principal. 

Hence,  to  find  the  principal, 

Cast  the  interest  on  one  dollar  for  the  given  time  and  divide 
the  given  interest  by  the  interest  so  found,  and  the  quotient  will 
he  the  principal. 

Quest. — ^218.  How  many  things  are  considered  in  all  questions  relating 
to  interest?  How  many  of  these  must  be  given  before  the  remaining  ones 
can  be  determined?  219,  What  are  given  m  Case  I.?  What  required? 
220.  What  are  given  in  Case  II.  ?    What  required  ?    How  do  you  find  th© 

principal  ? 


242  SIMPLE    INTEREST. 

EXAMPLES. 

1.  The  interest  on  a  certain  sum  for  1  year  and  4  months, 
at  6  per  cent,  is  $3007,7136 :  what  is  the  principal? 

The  interest  on  $1  for  the  same  time  is  $0,08.     Hence, 
$3007,7136  -^  0,08  =  $37596,42  =  principal. 

2.  The  interest  on  a  certain  sum  for  nine  months,  at  6  per 
cent,  is  $178,9582  :  what  is  the  principal  ? 

*     3.  The  interest  for  29  days  is  $2,78,  at  6  per  cent :  what 
is  the  principal  ? 

4.  The  interest  for  17  days,  at  6  per  cent,  is  $4,0366  : 
what  is  the  principal  ? 

5.  The  interest  on  a  certain  sum  for  1  year  1  month  and 
6  days,  at  7  per  cent,  is  $26,7381  :  what  is  the  sum?  If 
the  interest  for  the  same  time  be  $22,9184  at  the  rate  of  6 
per  cent,  what  will  be  the  sum  ?  For  the  same  time,  what 
will  be  the  principal,  when  the  rate  is  4  per  cent  and  in- 
terest $15,2790  ?  When  the  rate  is  5  per  cent  and  interest 
$19,0987  ?  When  the  interest  is  $21,0086  and  rate  5i  per 
cent  ?  When  the  rate  is  7^  per  cent  and  interest  $28,6479  ? 
When  the  rate  is  8  per  cent  and  interest  $30,5578  ? 


221.  Having  given  the  interest,  the  principal,  and  time,  to 
find  the  rate  per  cent  of  interest. 

If  interest  be  cast  at  different  rates,  on  the  same  sum  and 
for  the  same  time,  the  amounts  of  interest  will  be  propor- 
tional to  the  rates.     Therefore,  cast  the  interest  on  the  prin- 
cipal for  the  given  time,  at  1  per  cent  per  annum ;  then. 
Interest  at  1  per  cent  :  given  interest  :  :   1  pei  cent  :  rate. 

Hence,  to  find  the  rate  of  interest. 

Cast  the  interest  on  the  principal  for  the  given  tiine  at  1  per 
cent :  then  divide  the  given  interest  by  the  interest  so  found,  and 
the  quotient  will  he  the  rate  of  interest. 

Quest. — 221.  What  are  given  m  Case  III  ?  What  are  required  ?  How 
do  you  find  the  rate  of  interest  ? 

H 


SIMPLE    INTEREST.  243 

EXAMPLES. 

1.  The  interest  on  $437,21  for  9  years  and  9  months  is 
$127,8840  :  what  is  the  rate  of  interest? 

Interest  on  $437,21  for  9  years  and  9  monflis,  at  1  per 
cent,  is  $42,6280  :  hence, 

$127,8840  -^  42,6280  =  3  per  cent,  the  rate. 

2.  The  interest  on  $987,99,  for  5  years  2  months  and  9 
days,  is  $256,4657  :  what  is  the  rate  of  interest? 

Note. — In  examples  similar  to  the  above,  and  to  those  of  the 
following  section,  the  fractions  of  a  per  cent  less  than  a  quarter,  or 
of  a  day,  may  be  omitted.  Such  small  fractions  may  arise  from 
the  different  methods  of  computation. 


222.  Having  given  the  principal,  the  interest,  and  the  rale 
of  interest,  to  find  the  time. 

If  interest  be  cast  at  the  same  rate 'and  on  the  same  prm- 
cipal  for  different  times,  the  amounts  of  the  interest  will  be 
proportional  to  the  times.  Hence,  if  the  interest  on  the 
principal  be  cast  for  1  year,  we  shall  have, 

Interest  for  1  year  :  given  Interest  :  :   1  year  :  time. 

Hence,  to  find  the  time, 

Cast  the  interest  on  the  given  principal  at  the  given  rate  for 
one  year:  then  divide  the  given  interest  by  the  interest  so 
found ^  and  the  quotient  will  he  the  time. 

EXAMPLES. 

1.  The  interest  on  $15000,  at  7  per  cent  per  annum,  is 
$700  :  what  is  the  time  ? 

Interest  on  $15000  for  1  year  at  7  per  cent  =  $1050: 
hence,       -j3_o_o_  _  _^o_  _.  |  ^f  g,  year  =:  8  months. 

2.  The  interest  on  $1119,48,  at  7  per  cent  per  annum,  is 
$195,909  .   what  is  the  time  ? 

Quest.— 2ii2.  What  are  given  in  Case  IV.?  What  are  required?  How 
do  you  find  the  time  ?     • 


244 


REDUCTION    OF    CURRENCIES. 


A  Table,  showing  the  number  of  shillings  in  a  dollar  in 
each  State,  and  the  rate  of  interest :  also,  the  value  of  a 
dollar  expressed  in  parts  of  a  pound,  which  is  found  by- 
dividing  the  number  of  pence  in  a  dollar  by  the  number  in  a 
pound. 


States. 

No.  of  shillings 

Value  of  the  dollar  in 

Legal  rate  of 

to  the  dollar. 

pounds. 

interest. 

1 

Maine 

6  shillings 

Ufei  —  jy  12  ^3 

*J>  1 •*' 2 4  0 — '*'To" 

6  per  cent. 

2 

N.  Hampshire 

6  shillings 

$l=£J,%=£j\ 

6  per  cent. 

3 

Vermont 

6  shillings 

$l=.£J,%=£j\ 

6  per  cent. 

4 

Massachusetts 

6  shillings 

$l=£^Y^^£j\ 

6  per  cent. 

5 

Rhode  Island 

6  shillings 

$l=£Jf^=£^ 

6  per  cent. 

6 

Connecticut 

6  shillings 

$l=£J^^^£j\ 

6  per  cent. 

7 

New  York 

8  shillings 

$l=.£^%%=£i 

7  per  cent. 

8 

Ohio 

8  shillings 

$l=£^«^^=£i 

6  per  cent. 

9 

New  Jersey 

7^.  6d. 

$l=£^\\=£i 

6  per  cent. 

10 

Pennsylvania 

7s.  6d. 

6  per  cent. 

11 

Delaware 

7s.  6d. 

dhl ^90  -/:»  3 

$l=£J^\=£i 

6  per  cent. 

12 

Maryland 

7s.  6d. 

6  per  cent. 

13 

Michigan 

8  shillings 

$i=^:\'V=£f 

7  per  cent. 

14 

Indiana 

6  shillings 

$l=£J,\=£^ 

6  per  cent. 

15 

Illinois 

6  shillings 

$l=£^\\=^£J^ 

6  per  cent. 

16 

Missouri 

6  shillings 

$l=£J,%=£j\ 

6  per  cent. 

17 

Virginia 

6  shillings 

$1=£^Y,=£t\ 

6  per  cent. 

18 

Kentucky 

6  shillings 

tl^£if^^£^ 

6  per  cent. 

19 

Tennessee 

6  shillings 

$l=£i^%=£j\ 

6  per  cent. 

20 

North  Carolina 

10  shillings 

$l^£hn=£i, 

6  per  cent. 

21 

South  Carolina 

4s.  Sd. 

$l=£i%=£^\ 

7  per  cent. 

22 

Georgia 

As.  Sd. 

$l=£J^,=£,\ 

7  per  cent. 

23 

Alabama 

Fed.  money 

8  per  cent. 

24 

Mississippi 

6  shillings 

$l=£Jj%^£^ 

6  per  cent. 

25 

Louisiana 

Fed.  money 

6  per  cent. 

26 

Arkansas 

Fed.  money 

6  per  cent. 

27 

Florida 

Fed.  money 

6  per  cent. 

28 

Texas 

6  shillings 

$l=£i/^=£^ 

8  per  cent. 

29 

i  Nov.  Scotia  > 
\  and  Canada  5 

5  shillings 

$l=£^\\=£i 

6  per  cent. 

REDUCTION  OF  CURREINCIES. 


It  has  already  been  shown  (Art.  16),  that  Federal  Money 
is  the  currency  of  the  United  States  ;  the  pound,  however,  is 

occasionally  used.  • 


REDUCTION  OF  CURRENCIES.  245 

There  are  two  principal  reductions : 

1st.  To  change  any  sum  expressed  in  Federal  money  into 
pounds  shillings  and  pence. 

2d.  To  change  any  sum  expressed  in  poimds  shillings  and 
pence,  into  Federal  money. 

For  the  first, 

Multiply  the  sum  in  dollars  cents  and  mills,  hy  the  value  of 
$1  expressed  in  the  fraction  of  a  pound,  and  the  product  will 
be  the  corresponding  value  in  pounds  and  the  decimal  of  a  pound. 

For  the  second, 

Reduce  the  shillings  and  pence  to  the  decimal  of  a  pound  by 
Art.  161,  and  annex  the  decimal  to  the  entire  pounds.  Then 
multiply  by  the  fraction  with  its  terms  inverted,  which  expresses 
the  value  of  $1  in  terms  of  a  pound,  and  the  product  will  be 
dollars  cents  and  mills, 

EXAMPLES. 

1.  What  is  the  value  of  $375,87,  in  pounds  shillings  and 
pence,  New  York  Currency  ? 


We  first  multiply  by  f ,  and 
then  reduce  the  decimal  of  a 
pound  to  shillings  and  pence. 

2.  What  is  the  value  of  £127  18^.  6d,,  in  Federal  money, 
if  the  currency  be  6  shillings  to  the  dollar  ? 

We  first  reduce  the  shillings 
and  pence  to  the  fraction  of  a 
£,  and  then  multiply  by  the 
fraction  of  a  dollar  with  its 
terms  inverted. 

3.  What  is  the  value  of  $2863,75   in  pounds   shillings 
and  pence,  Pennsylvania  currency  ? 

4.  What  is  the  value  of  £459  3^.  6d.,  Georgia  currency, 
in  dollars  and  cents  ? 

5.  What  is  the  value  of  $9763,28,  in  pounds  shillings  and 
pence.  North  Carolina  currency  ? 

6.  What  is  the  value,  in  dollars  and  cents,  of  JB637'18j. 
8cZ.,  Nova  Scotia  currency? 


OPERATION. 

375,87  Xf =£150.348 
=£150  6s  U^d.+ 


OPERATION. 

£127  18;?.  6(f.=:  127.925 
127.925  xV»  =  $426,416+. 


246  COMPOUND    INTEREST. 


COMPOUND  INTEREST. 

223.  Compound  Interest  is  when  the  interest  on  a  sum  of 
money  becoming  due,  and  not  being  paid,  is  added  to  the 
principal,  and  the  interest  then  calculated  on  this  amount,  as 
on  a  new  principal.  For  example,  suppose  I  were  to  borrow 
of  Mr.  Wilson  $200  for  one  year,  at  6  per  cent.  If  at  the 
end  of  the  year  Mr.  Wilson  should  add  the  interest,  $12,  to 
the  principal,  $200,  making  $212,  and  charge  interest  on 
this  sum  till  paid,  this  would  be  Compound  Interest,  because 
it  is  interest  upon  interest.     Hence, 

Calculate  the  interest  to  the  time  at  which  it  becomes  due  : 
then  add  it  to  the  principal  and  calculate  the  interest  on  the 
amount  as  on  a  new  principal :  add  the  interest  again  to  the 
principal  and  calculate  the  interest  as  before :  do  the  same  for 
all  the  times  at  which  payments  of  interest  become  due :  from 
the  last  result  subtract  the  principal,  and  the  remainder  will  be 
the  compound  interest. 

EXAMPLES. 

1.  What  will  be  the  compound  interest,  at  7  per  cent,  of 
$3750  for  4  years,  the  interest  being  added  yearly? 

$3750,000      principal  for  1st  year. 
$3750  X  .07  :=     262,500      interest    for  1st  year. 
4012,500      principal  for  2d     " 
$4012,50     X  .07  =     280,875      interest    for  2d     " 
4293,375      principal  for  3d     " 
$4293,375  X  .07  =     300,536+  interest    for  3d     " 
4593,911+  principal  for  4t.h    " 
$4593,911  X  .07  =:     321,573+ interest    for  4th    " 
4915,484+  amount  at  4  years. 
1st  principal     3750,000 
Amount  of  interest     $  1 1 65,484  + 

Quest. — ^223.  What  is  compound  Interest  ?  How  do  you  find  the  com- 
pound interest  on  any  sum  ? 


COMPOUND    INTEREST. 


247 


2.  If  the  interest  be  computed  annually,  what  will  be  the 
interest  on  $300  for  th>ee  years,  at  6  per  cent  1 

3.  What  will  be  the  compound  interest  on  $590,74,  at  6 
per  cent,  for  2  years,  the  interest  being  added  annually  1 

4.  What  will  be  the  compound  interest  on  $500  for  1  year, 
at  8  per  cent,  the  interest  being  computed  quarterly  ? 

5.  What  will  be  the  compound  interest  on  $3758,56  for  3 
years,  at  7  per  cent,  the  interest  being  added  every  6  months  ? 

6.  What  will  be  the  compound  interest  on  $95637,50  for 
7  years,  at  6  per  cent,  the  interest  being  added  annually  ? 

7.  What  will  be  the  compound  interest  on  $75439,75 
for  4  years,  at  4^  per  cent,  the  interest  being  added  an- 
nually ? 


A    TABLE, 


224.  Showing  the  interest  of  £1,  or  $1,  compounded  an- 
nually, for  any  number  of  years  not  exceeding  20. 


Years. 

3  per  cent. 

3i  per  cent. 

4  per  cent. 

4*  per  cent. 

5  per  cent.  6  per  cent. 

1 

.030000 

.035000 

.040000 

.045000 

.050000  .060000 

2 

.060900 

.071225 

.081600 

.092025 

. 102500 

. 123600 

3 

.092727 

.108718 

.124864 

.141166 

.157625 

.191016 

4 

.125509 

.147523 

. 169859 

.192519 

.215506 

.262477 

5 

.159274 

.187686 

.216653 

.246182 

.275282 

.338226 

6 

.194052 

.229255 

.265319 

.302260 

.340096 

.418519 

7 

.229874 

.272279 

.315932 

.260862 

.407100 

.503630 

8 

.266770 

.316809 

.368569 

.422100 

.477455 

.593848 

9 

.304773 

.362897 

.423312 

.486095 

.551328 

.689479 

10 

.343916 

.410599 

.480244 

.552969 

.628895 

.790848 

11 

.384234 

.459970 

.539454 

.622853 

.710339 

.898299 

12 

.425761 

.511069 

.601032 

.695881 

.795856 

1.012196 

13 

.468534 

.563956 

.665074 

.772196 

.885649 

1.132928 

14 

.512590 

.618695 

.731676 

.851945 

.979932 

1.260904 

15 

.557967 

.675349 

.800944 

.935282 

1.078928 

1.396558 

16 

.604706 

.733986 

.872981 

1.022370 

1.182875 

1.540352 

17 

.652848 

.794676 

.947900 

1.113377 

1.292018 

1.692773 

18 

.702433 

.857489 

.025817 

1.208479 

1.406619 

1.854339 

19 

.753506 

.922501 

. 106849 

1.307860 

1.526950 

2.025600 

20 

.806111 

.989789 

.191123 

1.411714 

1.653298 

2.207135 

1 

*11 


2^8  COMPOUND    INTEREST. 

We  will  now  explain  the  method  of  finding  the  compound 
interest  on  any  sum,  for  any  time,  by  means  of  the  above 
table. 

Take  from  the  table  the  interest  of  £1  or  $1  for  the  same 
time,  and  at  the  same  rate,  and  then  multiply  the  number  so 
found  by  the  principal,  and  the  product  will  be  the  compound 
interest. 

EXAMPLES. 

1.  What  will  be  the  compound  interest  on  $350  for  three 
years,  at  6  per  cent  per  annum,  the  interest  being  computed 
annually  ? 

Interest  from  the  table  on  $1  =  $0.191016  ; 
then,     $0.191016  X  350  ~  $66.8556. 

2.  What  will  be  the  compound  interest  on  $856,95  for  15 
years,  at  3J  per  cent  per  annum  ? 

3.  What  will  be  the  compound  interest  on  $9864,05  for  16 
years,  the  interest  being  computed  annually,  at  4  per  cent? 

4.  What  will  be  the  compound  interest  on  $1675,20  for  20 
years,  at  4i  per  cent,  the  interest  being  computed  annually? 

5.  What  will  be  the  compound  interest  on  $5463,25  for  17 
years,  at  5  per  cent,  the  interest  being  computed  annually? 

6.  What  will  be  the  compound  interest  on  $3769,75  for 
18  years,  at  3  per  cent,  the  interest  being  computed  an- 
nually ? 

7.  What  will  be  the  compound  interest  on  £24  17^.  6d, 
for  10  years,  at  4  per  cent,  the  interest  being  computed  an- 
nually ? 

8.  What  will  be  the  compound  interest  on  $9854,50  for 
12  years,  at  6  per  cent,  the  interest  being  computed  annually  ? 

Quest. — 224.  How  do  you  find  the  compound  interest  on  any  sura  by 
the  table  ? 


LOSS    AND    GAIN 


249 


LOSS  AND  GAIN. 

225.  Loss  and  Gain  is  a  rule  by  which  merchants  dis- 
cover the  amount  lost  or  gained  in  the  purchase  and  sale  of 
goods.  It  also  instructs  them  how  much  to  increase  or  di- 
minish the  price  of  their  goods,  so  as  to  make  or  lose  so 
•nuch  per  cent. 

EXAMPLES. 

1.  Bought  a  piece  of  cloth  containing  75y(?.  at  $5,25  per 
yard,  and  sold  it  at  $5,75  per  yard :  how  much  was  gained 
in  the  trade  1 

OPERATION. 


We  first  find  the  profit 
on  a  single  yard,  and  then 
the  profit  on  the  75  yards. 


$5,75  price  of  1  yard. 
$5,25  cost  of  1  yard. 
.  50cts.  profit  on  1  yard. 


yd. 

yd. 

cts. 

1     : 

75     : 

:     50 
75 

Ans. 


$37,50 


Ans.  $37,50. 


2.  A  merchant  bought  a  bale  of  goods  containing  125 
yards  for  $687,75,  and  sold  it  at  auction  for  $4,50  per  yard : 
how  much  did  he  lose  in  all,  and  how  much  per  yard  ? 

$687,75  =  cost  of  the  bale. 
562,50  =  price  of  125  yd.  at  $4,50  per  yd. 


$125,25  =  total  loss. 


125)$125,25(1,00,2. 


Ans. 


Total  loss  $125,25. 
Loss  on  each  yd.  $1,00,2. 


Quest. — 225.  What  is  the  rule  of  loss  and  gain  ? 


250  LOSS    AND    GAIN. 

2.  Bought  a  piece  of  calico  containing  50yd.  at  2^.  6d. 
per  yard:  what  must  it  be  sold  for  per  yard  to  gain 
£1  0^.  10^^.  ? 

507/d,  at  2^.  6d,  =  £6  5s, 

Profit     =  £1  Os,  lOd, 
It  must  sell  for  £7  5s.  lOd. 


50)£7  5s.  I0d.(2s.  Ud. 

Ans.  2s.  lid, 

3.  Bought  a  hogshead  of  brandy  at  $1,25  per  gallon,  and 
sold  it  for  $78 :  was  there  a  loss  or  gain  ? 

4.  A  merchant  purchased  3275  bushels  of  wheat  for  which 
he  paid  $3517,10,  but  finding  it  damaged  is  willing  to  lose 
10  per  cent :  what  must  it  sell  for  per  bushel  ? 

226.  In  the  sale  of  goods,  knowing  the  per  cent  of  gain, 
and  the  amount  received,  to  find  the  principal  or  cost. 

I  sold  a  parcel  of  goods  for  $195,50,  on  which  I  made  15 
per  cent :  what  did  they  cost  me  1 

It  is  evident  that  the  cost  added  to  15  hundredths  of  the 
cost  will  be  equal  to  what  the  goods  brought,  viz.,  $195,50. 
If  we  call  the  cost  1,  then  1  plus  -^-^q  of  the  cost  will  be  equal 
to  what  they  bring  :  that  is, 

or,  cost  equals  $195,50  X  100  ^  115  =  $170. 

Hence,  to  find  the  cost. 

Multiply  the  amount  by  100  and  divide  the  product  by  100 
flus  the  per  cent  of  gain,  and  the  quotient  will  be  the  cost, 

227.  When  there  is  a  loss,  we  have  the  following  method : 
If  I  sell  a  parcel  of  goods  for  $170,  by  which  I  lose  15 

per  cent,  what  did  they  cost  ? 

It  is  evident  that  the  cost,  less  15  per  cent,  that  is,  less  15 
hundredths  of  the  cost,  is  equal  to  $170.     Hence.  85  hun- 

QuEST. — 226.  Knowing  the  per  cent  of  gain  and  the  amount  received, 
how  do  you  find  the  cost  ?  227.  Knowing  the  per  cent  and  the  amount 
lost,  how  do  you  find  the  cost  ? 


LOSS    AND    GAIN.  251 

dredths  of  the  cost  is  equal  to  $170 ;  and  consequently,  the 

cost  is  equal  to 

$170  X  100  -^  85  —  $200  cost. 
Hence,  to  find  the  cost  when  there  is  a  loss, 
Multiply  the  amount  received  by  100  and  divide  the  product 

by  the  difference  between   100  and  the  per  cent  lost,  and  the 

quotient  will  be  the  cost, 

EXAMPLES. 

1.  Sold  cloth  at  $1,25  per  yard  and  lost  15  per  cent:  for 
what  should  I  have  sold  it  to  have  gained  12  per  cent? 

2.  Sold  cloth  at  $1,25  per  yard  and  lost  15  per  cent: 
what  per  cent  should  I  have  gained  had  I  sold  it  at  $1,6470^^ 
per  yard  ? 

3.  Sold  cloth  at  $1,6470^^  per  yard  and  gained  12  per 
cent:  for  what  ought  I  to  have  sold  it  to  lose  15  per  cent  J 

4.  A  bought  a  piece  of  cotton  containing  80  yards,  at  6 
cents  per  yard  ;  he  sold  it  for  7^  cents  per  yard  r  how  much 
did  he  gain,  and  how  much  per  cent  ? 

5.  Bought  a  piece  of  cloth  containing  150  yards  for  $650  : 
what  must  it  be  sold  for  per  yard,  in  order  to  gain  $300  ? 

6.  Bought  a  quantity  of  wine  at  $1,25  per  gallon,  but  it 
proves  to  be  bad  and  I  am  obliged  to  sell  it  at  15  per  cent 
less  than  I  gave  :  how^  much  must  I  sell  it  for  per  gallon  ? 

7.  A  farmer  sells  375  bushels  of  corn  for  75cts.  per  bushel : 
the  purchaser  sells  it  at  an  advance  of  20  per  cent :  how 
much  did  he  receive  for  the  corn  1 

8.  A  merchant  buys  one  tun  of  wine  for  which  he  pays 
$725,  and  wishes  to  sell  it  by  the  hogshead  at  an  advance 
of  20  per  cent :  what  must  he  charge  per  hogshead  ? 

9.  A  merchant  buys  316  yards  of  calico  for  which  he  pays 
20  cents  per  yard  ;  one-half  is  so  damaged  that  he  is  obliged 
to  sell  it  at  a  loss  of  6  per  cent ;  the  remainder  he  sells  at 
an  advance  of  19  per  cent :  how  much  did  he  gain  ? 


252  STOCKS    AND    CORPORATIONS. 

10.  If  I  buy  coffee  at  16  cents  and  sell  it  at  20  cents,  how 
much  do  I  make  per  cent  on  the  money  paid  ? 

11.  If  I  buy  tea  at  4^.  per  pound  and  sell  it  at  4^.  9d.  per 
pound,  how  much  should  I  gain  on  a  purchase  of  £100  ? 

12.  A  merchant  bought  650  pounds  of  cheese  at  ;L0  cents 
per  pound,  and  sold  it  at  12  cents  per  pound :  how  much  did 
he  gain  on  the  whole,  and  how  much  per  cent  on  the  money 
laid  out  1 

13.  Bought  cloth  at  $2,50  per  yard,  which  proving  bad,  1 
wish  to  sell  it  at  a  loss  of  18  per  cent :  how  much  must  I  ask 
per  yard  ? 

14.  Bought  150  gallons  of  molasses  at  75  cents  a  gallon, 
30  gallons  of  which  leaked  out.  At  what  price  per  gallon 
must  the  remainder  be  sold  that  I  may  clear  10  per  cent  on 
the  cost? 


STOCKS  AND  CORPORATIONS. 

228.  Stock  is  a  general  name  for  the  money  contributed 
by  individuals  for  the  establishment  of  banks  and  manufac- 
turing companies,  and  the  individuals  who  contribute  the 
money  are  called  Stockholders. 

229.  The  individuals  so  associated  are  called,  in  their 
collective  capacity,  a  Corporation  ;  and  the  law  which  defines 
their  rights  and  powers,  is  called  the  Charter  of  the  Bank  or 
Company. 

230.  The  amount  of  money  paid  in  by  the  stockholders  to 
carry  on  the  business  of  the  corporation,  is  called  the  Capi- 
tal, The  capital  is  generally  divided  into  a  certain  numbei 
of  equal  parts  called  shares,  and  the  written  evidences  of 
ownership  of  such  shares,  are  called  certificates  of  stock.      • 

Quest. — ^228.  What  is  stock  ?  What  are  individuals  called  who  owu 
the  stock?  229.  What  are  they  called  in  their  associated  capacity?  What 
IS  the  law  called  which  incorporates  them  ?  230.  What  is  the  amount  of 
money  paid  in  by  the  stockholders  called  ?  How  is  the  capital  generally 
divided  1    What  is  the  evidence  of  ownership  called  ? 


COMMISSION    AND    BROKERAGE.  253 

231.  When  the  General  Government,  or  any  of  the  states 
borrows  money  for  public  purposes,  an  eviiience  is  given  to 
the  lender  in  the  form  of  a  bond,  bearing  a  given  interest. 
Such  bonds,  when  given  by  the  United  States,  are  called 
United  States  Stock ;  and  when  given  by  any  one  of  the 
states,  are  called  State  Stocks.  These  bonds  or  stocks  are 
generally  made  transferable  from  one  person  to  another. 

232.  The  nominal  or  par  value  of  a  stock  is  its  original 
cost;  that  is,  the  amount  named  in  the  certificate  or  bond. 
The  market  value  is  what  it  will  bring  when  sold.  If  the 
market  value  is  above  the  par  value,  the  stock  is  said  to  be 
at  a  premium,  or  above  par ;  but  if  the  market  value  is  below 
the  par  value,  it  is  then  said  to  be  at  a  discount,  or  below  par. 
For  example,  if  ^100  of  stock  will  bring  in  the  market  $110, 
the  stock  is  10  per  cent  above  par ;  if,  on  the  contrary,  it  will 
bring  but  $90,  it  is  10  per  cent  below  par:  the  percentage 
of  premium  or  discount  being  always  estimated  on  the  par 
value. 


COMMISSION  AND  BROKERAGE. 

233.  A  person  who  buys  or  sells  goods  for  another,  re- 
ceiving therefor  a  certain  rate  per  cent,  is  called  a  factor  or 
commission  merchant ;  and  the  percentage  on  any  purchase 
or  sale,  is  called  the  commission, 

234.  Dealers  in  money  or  stocks  are  called  Brokers,  and 
the  amount  of  their  commissions  on  any  purchase  or  sale,  is 
called  the  brokerage.  The  commission  for  goods  or  moneys 
is  generally  a  certain  per  cent  or  rate  per  hundred  on  the 
moneys  paid  out  or  received,  and  the  amount  may  be  deter- 
mined by  the  rules  of  simple  interest. 

Quest. — 231.  What  is  United  States  stock?  What  are  state  stocks? 
232.  Wiiat  is  the  nominal  or  par  value  of  a  stock  ?  What  is  the  market 
value  ?  What  do  you  understand  by  a  stock's  being  at  a  premium  ?  Whal 
by  its  being  at  a  discount?  233.  Wliat  is  the  business  of  a  commission 
merchant?  234.  What  is  the  business  of  a  broker?  How  is  the  com- 
mission on  goods  and  moneys  generally  estimated  '^ 


254 


COMMISSION    AND    BROKERAGE. 


The  commission  for  the  purchase  and  sale  of  goods  varies 
from  2i  to  12^  per  cent,  and  under  some  circumstances  even 
higher  rates  are  paid.  The  brokerage  on  the  purchase  and 
sale  of  stocks  in  Wall-street,  in  the  city  of  New  York,  is 
generally  one -fourth  per  cent  on  the  par  value  of  the  stock. 


.06 


EXAMPLES. 

1.  What  is  the  commission  on  $4396  at  6  per  cent^ 
We  here   find  the  com-  operation. 

mission,   as   in    simple    in- 
terest, by  multiplying  by  the  

decimal    which    expresses  ^263,76 

the  rate  per  cent.  ^ns.  $263,76. 

2.  A  factor  sells  120  bales  of  cotton  at  $425  per  bale,  and 
is  to  receive  2^  per  cent  commission  :  how  much  must  he 
pay  over  to  his  principaH 

3.  A  sent  to  B,  a  broker,  $3825  to  be  invested  in  stock: 
B  is  to  receive  2  per  cent  on  the  amount  paid  for  the  stock  • 
what  was  the  value  of  the  stock  purchased  ? 

OPERATION. 


As  B  is  to  receive  2  per 
cent,  it  follows  that  $102 
of  A's  money  will  purchase 
but  $100  of  stock:  hence, 
100-}-  the  commission,  is  to 
100,  as  the  given  sum  to 
the  value  of  the  stock  which 
it  will  purchase.  Hence, 
to  find  the  value  of  the  stock 
purchased. 


100 

2 

102 


100 


3825  :  Ans 
100 


102)382500(3750 
306 

765 
714 


510 
510 


Ans.  $3750. 


Multiply  the  amount  to  he  invested  hy  100  and  divide  the 
product  by  100  plus  the  brokerage. 

Quest  — What  is  the  general  commission  on  the  purchase  and  sale  of 
goods?  How  may  it  be  determined?  What  is  the  customary  broker- 
age on  the  purcliase  and  sale  of  stocks  ? 


COMMISSION  AND  BROKERAGE  255 


Amount  paid $3750 

Brokerage  on  $3750,  at  2  per  cent  =:        75 
Total  sum     -     -     $3825 


4.  I  have  $5000  to  be  laid  out  in  stocks  which  are  15  per 
cent  below  par :  how  much  will  it  purchase  at  the  par  or 
nominal  value  ? 

It  is  plain  that  every  85  dollars  will  purchase  stock  of  the 
par  value  of  $100  :  hence, 

$85   :   $100  :  :   $5000  :  Ans. 

Therefore,  to  find  how  much  will  be  purchased  at  the  par 
value,  when  the  stock  is  below  par. 

Multiply  the  sum  to  he  invested  hy  100  and  divide  the 
product  by  100  minus  the  discount. 

5.  A  person 'has  $7000  which  he  wishes  to  invest :  what 
will  it  purchase  in  5  per  cent  stocks,  at  3|-  per  cent  below 
par,  if  he  pays  \  per  cent  brokerage  ? 

6.  How  much  6  per  cent  stock  can  be  purchased  for 
$8500,  at  81  per  cent  premium,  \  per  cent  being  paid  to 
the  broker  ? 

7.  A  factor  receives  $708,75,  and  is  directed  to  purchase 
iron  at  $45  per  ton  :  he  is  to  receive  5  per  cent  on  the  money 
paid  :  how  much  iron  can  he  purchase  ? 

8.  Messrs.  P,  W,  &  K  buy  200  shares  of  United  States 
stock  for  Mr.  A.  The  par  value  is  $1000  dollars  a  share, 
the  stock  is  at  a  premium  of  6i  per  cent,  and  their  broker- 
age is  one-fourth  per  cent.  How  much  must  A  pay  them 
for  his  stock  ? 

9.  Messrs.  P,  W,  &  K  receive  $28750  to  be  invested  in 
stock.  They  charge  -1-  per  cent  commission  on  the  amount 
paid  :  what  is  the  value  of  the  stock  purchased  ? 

10.  The  par  value  or  first  cost  of  257  shares  of  bank  stock 
was  $200  per  share  :  what  is  the  present  value,  if  the  stock 
is  at  a  premium  of  15  per  cent,  that  is,  15  per  cent  above 
par"? 


256  BANKING. 

11.  What  would  be  the  value  of  the  stock  named  in  the 
last  example,  if  it  were  at  a  discount  of  10  per  cent? 

12.  One  hundred  shares  of  United  States  Bank  stock  was 
worth  18i  per  cent  premium :  the  par  value  being  $200  per 
share,  what  was  the  value  of  the  100  shares? 

13.  A  bank  fails,  and  has  in  circulation  bills  to  the  amount 
of  $267581.  It  can  pay  9i  per  cent:  how  much  money  is 
there  on  hand  ? 

14.  Sixty-nine  shares  of  bank  stock,  of  which  the  par 
value  is  $125,  is  at  a  discount  of  8  per  cent:  what  is  its 
value  ? 

15.  My  commission  merchant  sells  goods  to  the  amount 
of  f  1000,  on  which  I  allow  him  a  commission  of  2  per  cent ; 
and  as  he  pays  over  before  the  money  becomes  due,  I  allow 
him  li  per  cent :  how  much  am  I  to  receive  ? 

16.  My  broker  receives  from  me  $2000  to  be  laid  out  in 
stocks :  what  will  be  the  value  of  my  stocks  after  allowing 
him  ^  per  cent  commission  ? 

17.  I  sold  $13921,60  worth  of  goods  for  a  merchant  at  a 
commission  of  2^  per  cent :  how  much  ought  I  to  pay  over 
to  my  principal  ? 

18.  I  remitted  to  my  agent  $14760  to  lay  out  in  the  pur- 
chase of  iron.  He  takes  3^  per  cent  on  the  whole  sum  for 
his  commission,  and  then  buys  iron  at  95  dollars  per  ton : 
how  much  does  he  purchase  ? 


BANKING. 


235.  Banks  are  corporations  created  by  law  for  the  pur- 
pose of  receiving  deposites,  loaning  money,  and  furnishing  a 
paper  circulation  represented  by  specie. 

The  notes  made  by  a  bank  circulate  as  money,  because 
they  are  payable  in  specie  on  presentation  at  the  bank.  They 
are  called  hank  notes,  or  hank  hills. 

Quest. — 235.  What  are  banks  ?  Why  do  tho  notes  of  a  bank  circulate 
as  money  ?     What  are  they  called  1 


BANKmo.  257 


236.  The  note  of  an  individual,  or  as  it  is  generally  called, 
a  promissory  note  or  note  of  hand,  is  a  positive  engagement, 
in  writing,  to  pay  a  given  sum  at  a  time  specified,  and  to  a 
person  named  in  the  note,  or  to  his  order,  or  sometimes  to  the 
bearer  at  large. 


FORMS    OF    NOTES. 

Negotiable  Note. 
No.  1. 

$25,50.  Providence,  May  1,  1846. 

For  value  received  I  promise  to  pay  on  deman^,  to  Abel 
Bond,  or  order,  twenty-five  dollars  and  fifty  cents. 

Reuben  Holmes. 


Note  Payable  to  Bearer, 
No.  2. 


$875,39.  St.  Louis,  May  1,  1845. 

For  value  received  I  promise  to  pay,  six  months  after 
date,  to  John  Johns,  or  bearer,  eight  hundred  and  seventy-five  dol- 
lars and  thirty-nine  cents. 

Pierce  Penny. 


Note  by  two  Persons. 
No.  3. 


$659,27.  Buffalo,  June  2,  1846. 

For  value  received  we,  jointly  and  severally,  promise  to 
pay  to  Richard  Ricks,  or  order,  on  demand,  six  hundred  and  fifty 
nine  dollars  and  twenty-seven  cents.  Enos  Allan. 

John  Allan. 


Note  Payable  at  a  Bank. 
No.  4. 


$20,25.  Chicago,  May  7,  1846. 

Sixty  days  after  date,  I  promise  to  pay  John  Anderson, 
or  order,  at  the  Bank  of  Commerce  in  the  city  of  New  York, 
twenty  dollars  and  twenty-five  cents,  for  value  received. 

Jesse  Stqkes. 

Quest. — ^236.  What  is  a  promissory  note  ? 


258  BANKING. 

Remarks  relating  to  Notes. 

1.  The  person  who  signs  a  note,  is  called  the  drawer  or  maker 
of  the  note  ;  thus  Reuben  Holmes  is  the  drawer  of  note  No.  1 . 

2.  The  person  who  has  the  rightful  possession  of  a  note,  is  called 
the  holder  of  the  note. 

3.  A  note  is  said  to  be  negotiable  when  it  is  made  payable  to 
A  B,  or  order,  who  is  called  the  payee,  (see  No.  I.)  Now,  if  Abel 
Bond,  to  whom  this  note  is  made  payable,  writes  his  name  on  the 
back  of  it,  he  is  said  to  endorse  the  note,  and  he  is  called  the  en- 
dorser ;  and  when  the  note  becomes  due,  the  holder  must  first  de- 
mand payment  of  the  maker,  Reuben  Holmes,  and  if  he  declines 
paying  it»  the  holder  may  then  require  payment  of  Abel  Bond,  the 
endorser. 

4.  If  the  note  is  made  payable  to  A  B,  or  bearer,  then  the  drawer 
alone  is  responsible,  and  he  must  pay  to  any  person  who  holds  the 
note. 

5.  The  time  at  which  a  note  is  to  be  paid  should  always  be 
named,  but  if  no  time  is  specified,  the  drawer  must  pay  when  re- 
quired to  do  so,  and  the  note  will  draw  interest  after  the  payment 
is  demanded. 

6.  When  a  note,  payable  at  a  future  day,  becomes  due,  it  will 
draw  interest,  though  no  mention  is  made  of  interest. 

7.  In  each  of  the  States  there  is  a  7'ate  of  interest  established  by 
law,  which  is  called  the  legal  interest,  and  when  no  rate  is  speci- 
fied, the  note  will  always  draw  legal  interest.  If  a  rate  higher  than 
legal  interest  be  taken,  the  drawer,  in  most  of  the  States,  is  not 
bound  to  pay  the  note. 

8.  If  two  persons  jointly  and  severally  give  their  note,  (see 
No.  3,)  it  may  be  collected  of  either  of  them. 

9.  The  words  *'  For  value  received,"  should  be  expressed  in 
every  note. 

Quest. — 1.  What  is  the  person  called  who  signs  a  note?  2.  What  is 
the  person  called  who  owns  it  ?  3.  When  is  a  note  said  to  be  negotiable  ? 
What  is  the  person  called  to  whom  a  note  is  made  payable  ?  When  the 
payee  writes  his  name  on  the  back,  what  is  he  said  to  do  ?  What  is  he 
then  called  ?  4.  If  a  note  is  made  payable  to  A  B,  who  is  responsible  for  its 
payment  ?  5.  If  no  time  is  specified,  when  is  a  note  to  be  paid  1  6.  Will 
a  note  draw  interest  after  it  falls  due,  if  not  stated  in  the  note  ?  7.  If  the 
rate  of  interest  named  in  a  note  is  higher  than  the  legal  rate,  can  the 
amount  of  the  note  be  collected  ?  8.  If  two  persons  jointly  and  severally 
give  a  note,  of  whom  may  it  be  collected  ?  9.  What  words  should  be  put 
in  eveiy  note  ? 


BANK   DISCOUNT.  259 

10.  When  a  note  is  given,  payable  on  a  fixed  day,  and  in  a  spe- 
cific article,  as  in  wheat  or  rye,  payment  must  be  offered,  at  the 
specified  time,  and  if  it  is  not,  the  holder  can  demand,  the  value  in 
money. 

237.  By  mercantile  usage  and  the  custom  of  banks,  a  note 
does  not  really  fall  due  until  the  expiration  of  3  days  after 
the  time  mentioned  on  its  face".  For  example,  Note  No.  1 
would  be  due  on  the  4th  of  November,  and  the  three  ad- 
ditional days  are  called  days  of  grace'. 

When  the  last  day  of  grace  happens  to  be  a  Sunday,  or  a 
holiday,  such  as  New  Year's  or  the  4th  of  July,  the  note 
must  be  paid  the  day  before ;  that  is,  on  the  second  day  of 
grace. 


BANK  DISCOUNT. 

238,  Bank  Discount  is  the  charge  made  by  a  bank  for 
the  payment  of  money  on  a  note  before  it  becomes  due.  By 
the  custom  of  banks,  this  discount  is  the  interest  on  the 
amount  named  in  the  note,  from  the  time  the  note  is  dis- 
counted to  the  time  when  it  falls  due,  in  which  time  the  three 
days  of  grace  are  always  included.  The  amount  named  in 
a  note  is  called  the  face  of  it. 

The  PRESENT  VALUE  of  a  note  is  the  difference  between 
the  face  of  the  note  and  the  discount. 

239.  There  are  two  kinds  of  notes  discounted  at  banks : 
1st.  Notes  given  by  one  individual  to  another  for  property 
actually  sold — these  are  called  business  notes ^  or  business 
paper,    2d.  Notes  made  for  the  purpose  of  borrowing  money, 

Quest. — 10.  If  a  note  is  made  payable  on  a  fixed  day  and  in  a  specified 
article,  and  is  not  paid,  what  may  be  done?  237.  How  long  is  the  time 
for  the  payment  of  a  note  extended  by  mercantile  usage  ?  What  are  these 
days  called?  When  the  last  day  of  grace  falls  on  a  Sunday,  or  hofiday, 
when  must  the  note  be  paid?  238.  Wiiat  is  bank  discount?  How  is  it 
estimated?  How  is  it  estimated  by  the  custom  of  banks?  What  is  the 
face  of  a  note  ?  What  is  the  present  value  of  a  note  ?  239.  How  many 
kinds  of  notes  are  discounted  at  banks  ?  What  distinguislies  one  kind  from 
the  other,  and  what  are  they  called  ? 


260  BANK    DISCOUNT. 

which  are  called  accommodation  notes^  or  accommodation  paper. 

The  first  class  of  paper  is  much  preferred  by  the  banks,  as 

more  likely  to  be  paid  when  it  falls  due,  or  in  mercantile 

phrase,  *'  when  it  comes  to  maturity." 

Hence,  to  find  the  bank  discount  on  a  note. 
Add  "i  days  to  the  tihie  which  the  note  has  to  run  hefore  it 
.  hecomes  due,  and  calculate  the  interest  for  this  time  at  the  given 

rate  per  cent, 

EXAMPLES. 

1.  What  is  the  bank  discount  of  a  note  of  $1000  payable 
in  60  days,  at  6  per  cent  interest  ?  This  note  will  have  63 
days  to  run. 

2*.  A  merchant  sold  a  cargo  of  cotton  for  $15720,  for  which 
he  receives  a  note  at  6  months  :  how  much  money  will  he  re- 
ceive at  a  bank  for  this  note,  discounting  it  at  6  per  cent 
interest  1 

3.  What  is  the  bank  discount  on  a  note  of  $556,27  pay- 
able in  60  days,  discounted  at  6  per  cent  per  annum  1 

4.  A  has  a  note  against  B  for  $3456,  payable  in  th'tee 
months ;  he  gets  it  discounted  at  7  per  cent  interest :  how 
much  does  he  receive  1 

5.  What  is  the  bank  discount  on  a  note  of  $367,47,  having 
1  year,  1  month,  and  13  days  to  run,  as  shown  by  the  face 
of  the  note,  discounted  at  7  per  cent  ? 

6.  For  value  received  I  promise  to  pay  to  John  Jones,  four 
months  from  the  1 7th  of  July  next,  six  thousand  five  hundred 
and  seventy-nine  dollars  and  15  cents.  What  will  be  the  dis- 
count on  this,  if  discounted  on  the  1st  of  August,  at  6  per 
cent  per  annum  ? 

240.  It  is  often  necessary  to  make  a  note,  of  which  the 
present  value  shall  be  a  given  amount.  For  example,  if  I 
wish  to  receive  at  bank  the  sum  of  two  hundred  dollars,  for 
what  amount  must  I  give  my  note  payable  in  three  months  ? 

Quest. — Which  kind  is  preferred  ?  How  do  you  find  the  bank  discount 
on  a  note  ?     240.  What  is  often  necessary  in  bank  business  ? 


BANK    DISCOUNT.  261 

If  we  calculate  the  interest  on  one  dollar  for  the  time, 
which  will  be  3  months  added  to  the  3  days  of  grace,  and  at 
the  same  rate  per  cent,  this  will  be  the  bank  discount  on  $1 
payable  in  3  months  ;  and  if  this  discount  be  subtracted  from 
one  dollar,  the  remainder  will  be  the  present  value  of  one  dol- 
lar, to  be  paid  at  the  end  of  3  months.  Therefore, 
Pres.  val.  of  $1   :  pres.  val.  of  note  :   :  $1   :  amount  of  note. 

Hence,  to  find  the  face  of  a  note,  due  at  a  future  time  and 
bearing  a  given  interest,  that  shall  have  a  known  present  value, 

Find  the  present  value  of  $1  for  the  same  time  and  at  the 
same  rate  of  interest,  by  which  divide  the  present  value  of  the 
note,  and  the  quotient  will  be  the  face  of  the  note* 

EXAMPLES. 

1 .  For  what  sum  must  a  note  be  drawn  at  3  months,  so 
that  when  discounted  at  a  bank,  at  6  per  cent,  the  amount 
received  shall  be  $500  1 

Interest  on  $1  for  the  time,  3?no.  and  3c?a.=:$0,0155,  which 
taken  from  $1,  gives  present  value  of  $1  =  0,9845  ;  then 
$500  ^  0,9845  =  507,872  +  =  face  of  note. 

PROOF. 

Bank  interest  on  $507,872  for  3  months,  including  3  days 
of  grace,  at  6  per  cent  =  7,872,  which  being  taken  from  the 
face  of  the  note,  leaves  $500  for  its  present  value. 

2.  For  what  sum  must  a  note  be  drawn,  at  seven  per  cent, 
payable  on  its  face  in  1  year  6  months  and  14  days,  so  that 
when  discounted  at  bank  it  shall  produce  $307,27  ? 

3.  A  note  is  to  be  drawn  having  on  its  face  8  months  and 
12  days  to  run,  and  to  bear  an  interest  of  7  per  cent,  so  that 
it  will  pay  a  debt  of  $5450  :   what  is  the  amount  ? 

*  The  rule  founded  on  the  above  well-known  principle,  was,  it  is  believed, 
first  published  by  Roswell  C.  Smith,  in  his  New  Arithmetic. 

Quest. — What  will  be  the  presnnt  value  of  one  dollar  due  in  3  months  ? 
How  will  you  find  the  face  of  a  note,  of  a  given  present  value,  that  shall 
be  payable  at  a  future  time  ? 


262  DISCOUNT. 

4.  What  sum,  6  montlis  and  9  days  from  July  18th,  1846, 
drawing  an  interest  of  6  per  cent,  will  pay  a  debt  of  $674,89 
at  bank,  on  the  1st  of  August,  1846  ? 

5.  Mr.  Johnson  has  Mr.  Squires'  note  for  $874,57,  having 

4  months  to  run,  from  July  13th,  and  bearing  an  interest  of 

5  per  cent.  On  the  1st  of  October  he  wishes  to  pay  a  debt 
at  bank  of  $750,25,  and  gives  the  note  in  payment :  how 
much  must  he  receive  back  from  the  bank  ? 

6.  What  must  be  the  amount  of  a  note  discounted  at  6  pr  ct. 
having  4  months  and  7  days  to  run,  to  pay  a  debt  of  $1475,50  ! 

7.  Mr.  Jones,  on  the  1st  of  June,  desires  to  pay  a  debt  at 
bank  by  a  note  dated  May  16th,  having  6  months  to  run  and 
drawing  7  per  cent  interest :  for  what  amount  must  the  note 
be  drawn  the  debt  being  $1683,75  ? 

8.  What  amount  at  the  end  of  one  year,  with  grace,  in- 
terest at  5  per  cent,  will  pay  $1004,20  at  bank? 


DISCOUNT. 

# 

241.  If  I  give  my  note  to  Mr.  Wilson  for  $106,  payable  in 
one  year,  the  true  present  value  of  the  note  will  be  less  than 
$106  by  the  interest  on  its  present  value  for  one  year  ;  that  is, 
its  true  present  value  will  be' $100. 

The  true  present  value  of  a  note  is  that  sum  which  being 
put  at  interest  until  the  note  becomes  due,  would  increase  to 
an  amount  equal  to  the  face  of  the  note.  Thus,  $100  is  the 
true  present  value  of  the  note  to  Mr.  Wilson. 

The  discount  is  the  difference  between  the  face  of  a  note  and 
its  true  present  value.  Thus,  $6  is  the  discount  on  the  note 
to  Mr.  Wilson. 

To  find  the  true  present  value  of  a  note  due  at  a  future 
time,  find  the  interest  of  $1  for  the  same  time  ;  then, 

$1  -f-  its  interest  :  $1   :  •  given  sum.  :  its  present  value.  • 

Quest. — 241.  What  is  the  true  present  value  of  a  note?  What  is  the 
true  discount  ?  How  do  you  find  the  true  present  value  of  a  note  due  at  a 
future  time? 


DISCOUNT.  263 

Hence,  to  find  the  present  value  of  any  sum, 

Add  one  dollar  to  its  interest  for  the  given  time  and  divide 

the  given  amount  by  this  number,  and  the  quotient  will  be  the 

present  value. 

EXAMPLES. 

1.  What  is  the  present  value  of  a  note  for  $1828,75,  due 
in  one  year,  without  grace,  and  bearing  an  interest  of  4^  per 
cent  per  annum  1 

$1  +  its  interest  for  the  given  time  =  $1,045  : 
Hence,    $1828,75  -f-  $1,045  =:  $1750  the  present  value. 

PROOF. 

Int.  on  $1750  for  1  year,  at  4^  per  cent  =  $78,75 

Add  principal      - 1750 

Amount     -     -        $1828,75- 

2.  A  note  of  $1651,50  is  due  in  11  months,  without  grace, 
but  the  person  to  whom  it  is  payable  sells  it  with  the  dis- 
count off  at  7  per  cent :  how  much  shall  he  receive  1 

3.  How  much  ought  Mr.  Ready  to  pay  in  cash  for  his  note 
of  £36,  due  15  months  hence,  without  grace,  it  being  dis- 
counted at  5  per  cent  ? 

242.  Note. — When  payments  are  to  be  made  at  different  times, 
find  the  present  value  of  the  sums  separately,  and  their  sum  will 
he  the  present  value  of  the  note. 

4.  What  is  the  present  value  of  a  note  for  $10500,  on 
which  $900  are  to  be  paid  in  six  months ;  $2700  in  one 
year ;  $3900  in  eighteen  months ;  and  the  residue  at  the 
expiration  of  two  years,  all  without  grace,  the  rate  of  interest 
being  6  per  cent  per  annum  ? 

5.  What  is  the  discount  of  £4500,  one-half  payable  in  6 
months  and  the  other  half  at  the  expiration  of  a  year,  with- 
out grace,  at  7  per  cent  per  annum  1 


Quest. — 242.  When  payments  are  made  at  different  times,  how  do  yon 
find  the  true  present  value  ? 


264  DISCOUNT. 

6.  What  is  the  present  value  of  $5760,  one-half  payable 
in  3  months,  one-third  in  6  months,  and  the  rest  in  9  months, 
without  grace,  at  6  per  cent  per  annum  ? 

7.  Mr.  A  gives  his  note  to  B  for  $720,  one-half  payable 
in  4  months  and  the  other  half  in  8  months,  without  grace  : 
what  is  the  present  value  of  said  note,  discount  at  5  per  cent 
per  annum  ? 

8.  What  is  the  present  value  of  £825  payable  as  follows  • 
one-half  in  3  months,  one-third  in  6  months,  and  the  rest  in 
9  months,  without  grace,  the  discount  being  6  per  cent  per 
annum  ? 

9.  Bought  goods  for  £750  ready  money,  and  sold  them  for 
£900  payable  by  a  note  at  6  months,  without  grace  :  now,  if 
I  discount  the  note  at  6  per  cent  per  annum,  will  I  make 
or  lose  ? 

10.  What  is  the  present  value  of  $4000  payable  in  9 
months,  without  grace,  discount  4^  per  cent  per  annum  ? 

"11.  How  much  corn  must  I  carry  to  a  miller  that  I  may 
receive  a  bushel  of  meal,  -^  being  allowed  for  toll  and  waste  ? 

12.  Mr.  Johnson  has  a  note  against  Mr.  Williams  for 
$2146,50,  dated  August  17th,  1838,  which  becomes  due 
Jan.  11th,  1839  :  if  the  note  is  discounted  at  6  per  cent,  what 
ready  money  must  be  paid  for  it  September  25th,  1838  ? 

13.  C  owes  D  $3456,  to  be  paid  October  27th,  1842: 
C  wishes  to  pay  on  the  24th  of  August,  1838,  to  which  D 
consents  :  how  much  ought  D  to  receive,  interest  at  6  per 
cent? 

14.  What  is  the  present  value  of  a  note  of  $4800,  due  4 
years  hence,  without  grace,  the  interest  being  computed  at 
5  per  cent  per  annum  ? 

15.  A  man  having  a  horse  for  sale,  offered  it  for  $225 
cash  in  hand,  or  230  at  9  months,  without  grace  ;  the  buyer 
chose  the  latter  :  did  the  seller  lose  or  make  by  his  offer,  sup- 
posing money  to  be  worth  7  per  cent  1 


INSURANCE.  265 


INSURANCE. 

243.  Insurance  is  an  agreement,  generally  in  writing,  by 
which  an  individual  or  company  bind  themselves  to  exempt 
the  owners  of  certain  property,  such  as  ships,  goods,  houses, 
<fec.,  from  loss  or  hazard. 

The  written  agreement  made  by  the  parties,  is  called  the 
policy. 

The  amount  paid  by  him  who  owns  the  property  to  those 
who  insure  it,  as  a  compensation  for  their  risk,  is  called  the 
premium.  The  premium  is  generally  so  much  per  cent  on 
the  property  insured,  and  is  found  by  the  rules  for  simple 
interest. 

EXAMPLES. 

1.  What  would  be  the  premium  for  the  insurance  of  a 
house  valued  at  $8754  against  loss  by  jfire  for  1  year,  -dXj^ 
per  cent  1 

By  multiplying  by  .01,  we  have  the  insurance  (        ^^^  _  . 

at  1  per  cent (  ' 

The  half,  is  the  insurance  at  half  per  cent        -      $43,77. 

2.  What  would  be  the  premium  for  insuring  a  ship  and 
cargo,  valued  at  $147674,  from  New  York  to  Liverpool,  at 
3^  per  cent  ? 

3.  What  would  be  the  insurance  on  a  ship  valued  at 
$47520,  at  ^  per  cent  ?    Also  at  1  per  cent  ? 

4.  What  would  be  the  insurance  on  a  house  valued  at 
$16800,  at  li  per  cent?  Also  at  f  per  cent?  At  i  per 
cent  ?     At  ^  per  cent  ?     At  |  per  cent  ? 

5.  What  is  the  insurance  on  a  store  and  goods  valued  at 
$47000,  at  2\  per  cent  ?  At  2  per  cent ?  At  li  per  cent? 
At  f  per  cent  ?  At  i  per  cent  ?  At  -1-  per  cent  ?  At  ^  pet 
cent  1     At  1  per  cent  1 

Quest. — 243.  What  is  insurance?  What  is  the  written  agreement 
called  ?  What  is  the  amount  paid  for  the  insurance  called  ?  How  are  the 
premiums  generally  estimated  ?     How  are  they  found  ? 

12 


266  ASSESSING    TAXES. 

^  6.  A  merchant  wishes  to  insure  on  a  vessel  and  cargo  at 
sea,  valued  at  $28800  :  what  will  be  the  premium  at  1|^  per 
cent? 

7.  What  is  the  premium  on  f2250  at  1|  per  cent? 

8.  What  is  the  premium  on  $8750  at  3^  per  cent  ? 

9.  A  merchant  owns  three-fourths  of  a  ship  valued  at 
^24000,  and  insures  his  interest  at  2^  per  cent :  what  does 
he  pay  for  his  policy  ? 

10.  A  merchant  learns  that  his  vessel  and  cargo,  valued  at 
$36000,  have  been  injured  to  the  amount  of  $12000  ;  he 
effects  an  insurance  on  the  remainder  at  5J  per  cent :  what 
premium  does  he  pay  ? 

11.  What  is  the  insurance  on  my  house,  valued  at  $7500, 
at  ^  per  cent  ? 


ASSESSING  TAXES. 

244.  A  TAX  is  a  certain  sum  required  to  be  paid  by  the 
inhabitants  of  a  town,  county,  or  state,  for  the  support  of 
government.  It  is  generally  collected  from  each  individual, 
in  proportion  to  the  amount  of  his  property. 

In  some  states,  however,  every  white  male  citizen  over 
the  age  of  twenty -one  years  is  required  to  pay  a  certain  tax. 
This  tax  is  called  a  poll-tax ;  and  each  person  so  taxed  is 
called  a  poll. 

245.  In  assessing  taxes,  the  first  thing  to  be  done  is  to 
make  a  complete  inventory  of  all  the  property  in  the  town  on 
which  the  tax  is  to  be  laid.  If  there  is  a  poll  tax,  make  a 
full  list  of  the  polls  and  multiply  the  number  by  the  tax  on 
each  poll,  and  subtract  the  product  from  the  whole  tax  to  be 
raised  by  the  town ;  the  remainder  will  be  the  amount  to  be 
raised  on  the  property.     Having  done  this,  divide  the  whole 

Quest. — 244.  What  is  tax  ?  How  is  it  generally  collected  ?  What  is  a 
poll-tax?  245.  What  is  the  first  thing  to  be  done  in  assessing  a  tax?  If 
there  is  a  poll-tax,  how  do  you  find  the  amount  ?  How  then  do  you  find 
the  per  cent  of  tax  to  be  levied  on  a  dollar  1 


ASSESSING    TAXES.  267 

tax  to  be  raised  by  the  amount  of  taxable  property,  and  the 
quotient  will  be  the  tax  on  $1.  Then  multiply  this  quotient 
by  the  inventory  of  each  individual,  and  the  product  will  be 
the  tax  on  his  property. 

EXAMPLES. 

1.  A  certain  town  is  to  be  taxed  $4280;  the  property  on' 
which  the  tax  is  to  be  levied  is  valued  at  $1000000.     Now 
there  are  200  polls,  each  taxed   $1,40.     The  property  of 
A  is  valued  at  $2800,  and  he  pays  4  polls, 
B's  at  $2400,  pays  4  polls,  E's  at  $7242,  pays  4    polls, 

C's  at  $2530,  pays  2     "  F's  at  $1651,  pays  6       " 

D's  at  $2250,  pays  6     "  G's  at  $1600,80  pays  4   " 

What  will  be  the  tax  on  one  dollar,  and  what  will  be  A's 
tax,  and  also  that  of  each  on  the  list  1 
First,        $1,40  X  200  —  $280  amount  of  poll-tax. 

$4280  —  $280  =  $4000  amount  to  be  levied  on  property. 
Then,  $4000  ^  $1000000  =  4  mills  on  $1. 

Now,  to  find  the  tax  of  each,  as  A's,  for  example, 
A's  inventory     -       -       -       $2800 

,004 

11,20 

4  polls  at  $1,40  each     -  5,60 


A's  whole  tax   -       -       -       $16,80 


In  the  same  manner  the  tax  of  each  person  in  the  town- 
ship may  be  found. 

246.  Having  found  the  per  cent,  or  the  amount  to  be  raised 
on  each  dollar,  form  a  table  showing  the  amount  which  certain 
sums  would  produce  at  the  same  rate  per  cent.  Thus,  after 
having  found,  as  in  the  last  example,  that  four  mills  are  to  be 
raised  on  every  dollar,  we  can,  by  multiplying  in  succession 
by  the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  &c.,  form  the  following 

Quest. — How  do  you  then  find  the  amount  to  be  levied  on  each  in- 
dividual ?     246.  How  do  you  form  an  assessment  table  ? 


268 


ASSESSING    TAXES. 


$ 

,     $ 

$                $ 

$ 

$             1 

1  gives  0,004 

20  gives  0,080 

300  gives  1,200     | 

2 

"      0,008 

30      "      0,120 

400 

"      1,600 

3 

"      0,012 

40      "      0,160 

500 

'      2,000 

4 

'*      0,016 

50      "      0,200 

600 

'      2,400 

5 

^      0,020 

60      "      0,240 

700 

'      2,800 

6 

*      0,024 

70      "      0,280 

800 

'      3,200 

7 

'      0,028 

80      "      0,320 

900 

*      3,600 

8 

*       0,032 

90      "      0,360 

1000 

*      4,000 

9 

'      0,036 

100       "      0,400 

2000 

'      8,000 

10 

'      0,040 

200      "      0,800 

3000 

«    12,000 

This  table  shows  the  amount  to  be  raised  on  each  sum  in 
the  columns  under  $'s. 

1 .  To  find  the  amount  of  B's  tax  from  this  table. 

B's  tax  on  $2000      -     .     is     -     $8,000 

B's  tax  on       400      -     -     is     -     $1,600 

B's  tax  on  4  polls,  at  $1,40       -     $5,600 

B's  total  tax     - 


C's  tax  on  $2000      - 
C's  tax  on       500      - 
C's  tax  on         30 
C's  tax  on  2  polls 

C's  total  tax    - 

In  a  similar  manner,  we  might  find  the  taxes  to  be  paid 
by  D,  E,  &c. 

2.  In  a  county  embracing  350  polls,  the  amount  of  prop- 
erty on  the  tax  list  is  $318200  ;  the  amount  to  be  raised  is 
as  follows:  for  state  purposes  $1465,50;  for  county  pur- 
poses $350,25  ;  and  for  town  purposes  $200,25.  By  a  vote 
of  the  county,  a  tax  is  levied  on  each  poll  of  $1,50:  how 
liiuch  per  cent  will  be  laid  upon  the  property  1 

3.  In  a  county  embracing  a  population  of  98415  persons, 
a  tax  is  levied  for  town,  county,  and  state  purposes,  amount- 


is 

-  $15,200 

fron 

is 

is 

is 

is 

1  the  table. 

-  $8,000 

-  $2,000 

-  $0,120 

-  $2,800 

is 

-   $12,920 

EQUATION    OF    PAYMENTS  .  269 

ing  to  $100406.  Of  this  sum,  a  part  is  to  be  raised  by  a  tax 
of  25  cents  on  each  poll,  and  the  remainder  by  a  tax  of  two 
mills  on  the  dollar :  what  was  the  amount  of  property  ort  the 
tax  list  1 


EQUATION  OF  PAYMENTS. 

247.  1  OWE  Mr.  Wilson  $2  to  be  paid  in  6  months ;  $3  to 
be  paid  in  8  months  ;  and  $1  to  be  paid  in  12  months.  I 
wish  to  pay  his  entire  dues  at  a  single  payment,  to  be  made 
at  such  a  time,  that  neither  he  nor  I  shall  lose  interest:  at 
what  time  must  the  payment  be  made  ? 

The  method  of  finding  the  mean  time  of  payment  of 
several  sums  due  at  different  times,  is  called  Equation  of 
Payments. 

Taking  the  example  above, 
Int.  of  $2  for    6mo.  =  int.  of  $1  for  \2mo,     2  X     6  =  12 

"    of  $3  for    8mo.  =  int.  of  $1  for  2Amo,     3  X     8  =  24 

"    of  $1  for  \2mo.  =  int.  of  $1  for  l2mo.     1  X  12  =  12 
$6"  "48"  "is" 

The  interest  on  all  the  sums,  to  the  times  of  payment,  is 
equal  to  the  interest  of  $1  for  48  months.  But  48  is  equal 
to  the  sum  of  all  the  products  which  arise  from  multiply- 
ing each  sum  by  the  time  at  which  it  becomes  due  :  hence, 
the  sum  of  the  products  is  equal  to  the  time  which  would  be 
necessary  for  $1  to  produce  the  same  interest  as  would  be 
produced  by  all  the  sums. 

Now,  if  $1  will  produce  a  certain  interest  in  48  months, 
in  what  time  will  $6  (or  the  sum  of  the  payments)  produce 
the  same  interest  ?  The  time  is  obviously  found  by  dividing 
48  (the  sum  of  the  products)  by  $6,  (the  sum  of  the  pay- 
ments.) 

Quest. — 247.  What  is  Equation  of  Payments  ?  What  is  the  sum  of  the 
products,  which  arise  from  multiplying  each  payment  by  the  time  to  wliich 
it  becomes  due,  equal  to  ? 


200  X 
200  X 
200  X 

2  =  400 

4=  800 
6  z=z  1200 

6|00 

)24  00 
4 
Ans.   4  months 

270  •  EQUATION    OF    PAYMENTS. 

Hence,  to  find  the  mean  time, 

Multiply  each  payment  ly  the  time  before  it  becomes  due,  and 
divide  the  sum  of  the  products  by  the  sum  of  the  payments :  the 
juotient  will  be  the  mean  time. 

EXAMPLES. 

1.  B  owes  A  $600:  $200  is  to  be  paid  in  two  months, 
$200  in  four  months,  and  $200  in  six  months :  what  is  the 
mean  time  for  the  payment  of  the  whole  ? 

OPERATION. 

We  here  multiply  each  sum 
by  the  time  at  which  it  be- 
comes due,  and  divide  the  sum 
of  the  products  by  the  sum  of 
the  payments. 

2.  A  merchant  owes  $1200,  of  which  $200  is  to  be  paid 
in  4  months,  $400  in  10  months,  and  the  remainder  in  16 
months  :  if  he  pays  the  whole  at  once,  at  what  time  must  he 
make  the  payment  ? 

3.  A  merchant  owes  $1800  to  be  paid  in  12  months,  $2400 
to  be  paid  in  6  months,  and  $2700  to  be  paid  in  9  months : 
what  is  the  equated  time  of  payment  ? 

4.  A  owes  B  $2400  ;  one-third  is  to  be  paid  in  6  months, 
one  fourth  in  8  months,  and  the  remainder  in  12  months : 
what  is  the  mean  time  of  payment  ? 

5.  A  merchant  has  due  him  $600  to  be  paid  in  30  days, 
$1000  to  be  paid  in  60  days,  and  1500  to  be  paid  in  90  days : 
what  is  the  equated  time  for  the  payment  of  the  whole  ? 

6.  A  merchant  has  due  him  $4500 ;  one-sixth  is  to  be 
paid  in  4  months,  one-third  in  6  months,  and  the  rest  in  12 
months :  what  is  the  equated  time  for  the  payment  of  the 
whole  ? 

Quest. — How  do  you  find  the  mean  time  of  payment?  When  you 
reckon  the  time  from  the  date  at  which  the  first  payment  becomes  due,  do 
you  include  the  first  payment  ? 


EQUATION    OF    PAYMENTS.  271 

Note  1. — ^If  one  of  the  payments  is  due  on  the  day  from  which  the 
equated  time  is  reckoned,  its  corresponding  product  will  be  nothing, 
but  the  payment  must  still  be  added  in  finding  the  sum  of  the  pay- 
ments, 

7.  I  owe  $1000  to  be  paid  on  the  1st  of  January,  $1500 
on  the  1st  of  February,  $3000  on  the  1st  of  March,  and 
$4000  on  the  15th  of  April:  reckoning  from  the  1st  of 
January,  and  calling  February  28  days,  on  what  day  must 
the  money  be  paid  ? 

Note  2. — In  finding  the  equated  time  of  payments  for  several 
sums,  due  at  different  times,  any  day  may  be  assumed  as  the  one 
from  which  we  reckon.  Thus,  if  I  owe  Mr.  Wilson  Si 00  to  be 
paid  on  the  15th  of  July,  $208  on  the  15th  of  August,  and  $300  on 
the  9th  of  September,  and  we  require  the  mean  time  of  a  single 
payment,  it  would  be  most  convenient  to  estimate  from  the  1st  of 
July. 

From  1st  of  July  to  1st  payment  14  days 
"  "  "  to  2d  payment  45  days 
"         "         "     to  3d  payment  70  days. 

100  X  14  =     1400 


Then,  by  rule  ^iven  above,  we 
have, 


200  X  45  =    9000 
300  X  70  =:  21000 


600  6|00)314|0Q 

52i 


Hence,  the  amount  will  fall  due  in  52^  days  from  the  1st 
of  July  ;  that  is,  on  the  22d  day  of  August. 

But  we  may,  if  we  please,  demand  at  what  time  the  pay- 
ment would  be  due  from  the  1st  of  June. 

From  June  1st  to  1st  payment  44  days 
"  "  "  to  2d  payment  75  days 
"        "      "    to  3d  payment  100  days. 

Thus,     .  100  X    44  =    4400 

200  X     75  =  15000 
300  X  100  -  30000         ^ 
600  6|00)494|00 


272  EQUATION    OF    PAYMENTS. 

Hence,  the  payment  becomes  due  in  82^  days  from  June 
jLSt,  or  on  the  22d  of  August — the  same  as  before. 

Any  day  may^  therefore^  he  taken  as  the  one  from  which  the 
mean  time  is  estimated. 

9.  Mr.  Jones  purchased  of  Mr.  Wilson,  on  a  credit  of  six 
months,  goods  to  the  following  amounts  : 

15th  of  January,     a  bill  of  $3750, 

10th  of  February,  a  bill  of    3000, 

6th  of  March,       a  bill  of    2400, 

8th  of  June,  a  bill  of    2250. 

He  wishes,  on  the  1st  of  July,  to  give  his  note  for  the 

amount ;  at  what  time  must  it  be  made  payable  1 

10.  Mr.  Gilbert  bought  $4000  worth  of  goods:  he  was  to 
pay  $1600  in  five  months,  $1200  in  six  months,  and  the  re- 
mainder in  eight  months  :  what  will  be  the  time  of  credit,  if 
he  pays  the  whole  amount  at  a  single  payment  ? 

11.  A  owes  B  $1200,  of  which  $240  is  to  be  paid  in  three 
months,  $360  in  five  months,  and  the  remainder  in  ten  months 
what  is  the  mean  time  of  payment  ? 

12.  A  merchant  bought  several  lots  of  goods,  as  follows  : 

A  bill  of  $650,  June      6th, 
Do.      of    890,  July       8th, 
Do.      of  7940,  August  1st. 
Now,  if  the  credit  is  6  months,  at  what  time  will  the  whole 
become  due  ? 

13.  Mr.  Swain  bought  goods  to  the  amount  of  $3840,  to  be 
paid  for  as  follows,  viz.:  one-fourth  in  cash,  one-fourth  in  6 
months,  one-fourth  in  7  months,  and  the  remainder  in  one 
year :  what  is  the  average  time  of  payment  ? 

14.  Mr.  Johnson  sold,  on  a  credit  of  8  months,  the  follow- 
ing  bills  of  goods  : 

April  1st,  a  bill  of  $4350, 
'       May  7th,  a  bill  of    3750, 
June  5th,  a  bill  of    2550. 
At  what  time  will  the  whole  become  due  ? 


PARTNERSHIP    OR    FELLOWSHIP.  273 


PARTNERSHIP  OR  FELLOWSHIP. 

248.  Partnership  or  Fellowship  is  the  joining  together 
of  several  persons  in  trade,  with  an  agreement  to  share  the 
losses  and  profits  according  to  the  amount  which  each  one 
puts  into  the  partnership.  The  money  employed  is  called 
the  Capital  Stock. 

The  gain  or  loss  to  be  shared  is  called  the  Dividend, 
It  is  plain  that  the  whole  stock  which  suffers  the  gain  or 
loss,  must  be  to  the  gain  or  loss,  as  the  stock  of  any  individual 
to  his  part  of  the  gain  or  loss.     Hence, 

As  the  whole  stock  is  to  each  marl's  share,  so  is  the  whole 
gain  or  loss  to  each  marHs  share  of  the  gain  or  loss 

PROOF. 

Add  all  the  separate  profits  or  shares  together ;  their  sum 
should  be  equal  to  the  gross  profit  or  stock. 

EXAMPLES. 

1.  A  and  B  buy  certain  merchandise  amounting  to  £160, 
of  which  A  pays  £90,  and  B  £70  :  they  gain  by  the  purchase 
£32  :  what  is  each  one's  share  of  the  profits  ? 

A  -  -  £90 
B  -  -  £70 

-5Ta(\       S  90 )  ^oo       ^  £18  A's  share. 

£160  :    l^^.\    :  :  £32  :    i  £14  b's  share. 


2.  A  and  B  have  a  joint  stock  of  $4200,  of  which  A  owns 
$3600,  and  B  $600 :  they  gain  in  a  year  $2000 :  what  is 
each  one's  share  of  the  profits  ? 

3.  A,  B,  C,  and  D  have  £40,000  in  trade :  at  the  end  of 
six  months  their  profits  amount  to  £16,000:  what  is  each 
one's  share,  supposing  A  to  receive  £50  and  D  £30  out  of 
the  profits,  for  extra  services  ? 

Quest. — 248.  What  is  Partnership,  or  Fellowship  ?    What  is  the  gain  or 
loss  called?     What  is  the  rule  for  finding  each  one's  share? 
12* 


274  DOUBLE    FELLOWSHIP. 

4.  Five  persons,  A,  B,  C,  D,  and  E,  have  to  share  between 
them  an  estate  of  $20,000 :  A  is  to  have  one-fourth,  B  one- 
eighth,  C  one-sixth,  D  one-eighth,  and  E  what  is  left :  what 
will  be  the  share  of  each  ? 


DOUBLE  FELLOWSHIP. 

249.  When  several  persons  who  are  joined  together  in 
trade,  employ  their  capital  for  different  periods  of  time,  the 
partnership  is  called  Double  Fellowship. 

For  example,  suppose  A  puts  $100  in  trade  for  5  years, 
B  $200  for  2  years,  and  C  $300  for  1  year:  this  would 
make  a  case  of  double  fellowship. 

Now  it  is  plain  that  there  are  two  circumstances  which 
should  determine  each  one's  share  of  the  profits  :  1^^,  The 
amount  of  capital  he  puts  in ;  and  2dly,  The  time  which  it  is 
continued  in  the  business. 

Hence,  each  man's  share  should  be  proportional  to  the 
capital  he  puts  in,  multiplied  by  the  time  it  is  continued  in 
trade.     Therefore,  to  find  each  share, 

Multiply  each  man's  stock  by  the  time  he  continues  it  in 
trade ;  then  say,  as  the  sum  of  the  products  is  to  each  par- 
ticular product,  so  is  the  whole  gain  or  loss  to  each  man's  share 
of  the  gain  or  loss. 

EXAMPLES. 

1.  A  and  B  enter  into  partnership:  A  puts  in  £840  for  4 

months,  and  B  puts  in  £650  for  6  months  :  they  gain  £300  • 

what  is  each  one's  share  of  the  profits  ? 

A's  stock  £840x4  =  3360 

B's  stock  £650x6:^.3900  ^    ^-   ^• 

£^^.  U360>   ..  ^  U38  16  10 

£7260.  {^qqq}   ..£300.   <  jgi     3     j 


Quest. — 249.  What  is  Double  Fellowship?  What  two  circumstances 
determine  each  one's  share  of  the  profits  ?  Give  the  rule  for  finding  each 
one's  share. 


DOUBLE    FELLOWSHIP.  275 

2.  A  put  in  trade  £50  for  4  months,  and  B  £60  for  5 
months  :  they  gained  £24  :  how  is  it  to  be  divided  between 
them? 

3  C  and  D  hold  a  pasture  together,  for  which  they  pay 
£54:  C  pastures  23  horses  for  27  days,  and  D  21  horses 
for  39  days  :  how  much  of  the  rent  ought  each  one  to  pay  ? 

GENERAL  EXAMPLES  IN  FELLOWSHIP. 

1.  A  bankrupt  is  indebted  $2729,  viz.:  to  A  $509,37;  to 
B  $228 ;  to  C  $1291,23  ;  and  to  D  $709,40  ;  but  his  estate 
is  only  worth  $2046,75.  How  much  can  he  pay  on  the  dol- 
lar, and  how  much  will  each  creditor  receive  ? 

2.  A,  B,  and  C  send  a  ship  to  sea,  which  together  with 
her  cargo  was  worth  $15000.  A  and  B  owned  each  one- 
fifth,  and  C'the  rest.  They  gained  $1250:  how  much  did 
each  pay  towards  the  ship  and  cargo,  and  what  did  each  re- 
ceive of  the  profits  ? 

3.  A  man  bequeathed  his  estate  to  his  four  sons  in  the  fol- 
lowing manner,  viz. :  to  his  first  $5000  ;  to  his  second  $4500  ; 
to  his  third  $4500  ;  and  to  his  fourth  $4000.  But  on  settling 
his  estate,  it  was  found  that  after  paying  debts,  charges,  Slc, 
only  $12000  remained  to  be  divided:  how  much  must  each 
receive  ? 

4.  A  widow  and  her  two  sons  have  a  legacy  of  $4500,  of 
which  the  widow  is  to  have  one-half  and  the  sons  each  one- 
fourth.  Now  suppose  the  eldest  son  to  relinquish  his  share, 
and  the  whole  to  be  divided  in  the  above  proportions  between 
the  mother  and  youngest  son,  what  will  each  receive  ? 

5.  Suppose  premiums  to  the  value  of  $12  are  to  be  dis- 
tributed in  a  school  in  the  following  manner.  The  premiums 
are  divided  into  three  grades.  The  value  of  a  premium  of 
the  first  grade  is  twice  the  value  of  one  of  the  second ;  and 
the  value  of  one  of  the  second  grade  twice  that  of  the  third. 
Now  there  are-  6  to  receive  premiums  of  the  first  grade,  12 
of  the  second,  and  6  of  the  third :  what  will  be  the  value  of 
a  single  premium  of  each  grade  ? 


276  ALLIGATION    MEDIAL. 

6.  Four  traders  form  a  company :  A  puts  in  $300  for  5 
months,  B  $600  for  7  months,  C  $960  for  8  months,  D  $1200 
for  9  months.  In  the  course  of  trade  they  lost  $750  :  how 
much  falls  to  the  share  of  each  ? 

7.  A  and  B  lay  out  certain  sums  in  merchandise  amount- 
ing to  $320,  of  which  A  pays  $180  and  B  $140 ;  they  gain 
by  the  purchase  $64 :  what  is  each  one's  share  ? 


ALLIGATION  MEDIAL. 

250.  A  MERCHANT  mixcs  8lb.  of  tea,  worth  75cts.  per 
pound,  with  I6lb.  worth  $1,02  per  pound :  what  is  the  value 
of  the  mixture  per  pound  ? 

The  manner  of  finding  the  price  of  this  mixture  is  called 
Alligation  Medial.     Hence, 

Alligation  Medial  teaches  the  method  of  finding  the  price 
of  a  mixture  when  the  simples  of  which  it  is  composed,  and  their 
prices,  are  known. 

In  the  example  above,  the  simples  8lb.  and  I6lb.,  and  also 
their  prices  per  pound,  7dcts.  and  $1,02,  are  known. 

Sib.  of  tea  at  75cts.  per  lb. 6,00 

leib.     "      "     $1,02  per  Z6.      -       -       -       -       -       16,32 
24  sum  of  simples.  Total  cost  $22,32 


Now,  if  the  entire  cost  of  the  mix- 


24)22,32(93c^5. 
216 

72" 
72 


ture,  which  is  $22,32,  be  divided  by 
24,  the  number  of  pounds  or  sum  of 
the  simples,  the  quotient  93cts.  will  be 
the  price  per  pound.  Hence,  to  find 
the  price  of  the  mixture. 

Divide  the  entire  cost  of  the  whole  mixture  by  the  sum  of  the 
simples,  and  the  quotient  will  be  the  price  of  the  mixture. 

Quest. — 250.  What  is  Alligation  Medial  ?    How  do  you  find  the  price 
of  the  mixture  ? 


OPERATION. 


ALLIGATION    ALTERNATE.  277 


EXAMPLES. 


1.  A  farmer  mixes  30  bushels  of  wheat  worth  5s.  per 
bushel,  with  72  bushels  of  rye  at  3^.  per  bushel,  and  with 
60  bushels  of  barley  worth  2s.  per  bushel ;  what  is  the  value 
of  a  bushel  of  the  mixture  ? 


30  b 

lushe] 

Is  of  wheat  at  5s. 

-       -       150^. 

72 

it 

"  rye  at  3^. 

-       -       216^. 

60 
62 

(( 

"  barley  at  2^. 

■       -       120^. 

162)486(3^. 
486 

-*              Ans.  3s. 

2.  A  wine  merchant  mixes  15  gallons  of  wine  at  $1  per 
gallon  with  25  gallons  of  brandy  worth  75  cents  per  gallon  • 
what  is  the  value  of  a  gallon  of  the  compound  ? 

3.  A  grocer  mixes  80  gallons  of  whiskey  worth  Slcts.  per 
gallon  with  6  gallons  of  water,  which  costs  nothing :  what 
is  the  value  of  a  gallon  of  the  mixture  ? 

4.  A  goldsmith  melts  together  2lb.  of  gold  of  22  carats 
fine,  6oz.  of  20  carats  fine,  and  6oz.  of  16  carats  fine  :  what 
is  the  fineness  of  the  mixture  ? 

5.  On  a  certain  day  the  mercuiy  in  the  thermometer  was 
observed  to  average  the  following  heights  :  from  6  in  tho 
morning  to  9,  64°  ;  from  9  to  12,  74°  ;  from  12  to  3,  84^ ; 
and  from  3  to  6,  70^ :  what  was  the  mean  temperature  of 
the  day  ? 


ALLIGATION  ALTERNATE. 

251.  A  FARMER  would  mix  oats  worth  3,9.  per  bushel  with 
wheat  worth  9^.  per  bushel,  so  that-  the  mixture  shall  be 
worth  5s.  per  bushel :  what  proportion  must  be  taken  of  each 
sort? 

The  method  of  finding  how  much  of  each  sort  must  be 
taken  is  called  Alligation  Alternate.     Hence, 


278  ALLIGATION    ALTERNATE. 

Alligation  Alternate  teaches  the  method  of  finding  what 
proportion  must  he  taken  of  several  simples,  whose  prices  are 
known,  to  form  a  compound  of  a  given  price. 

Alligation  Alternate  is  the  reverse  of  Alligation  Medial, 
and  may  be  proved  by  it. 

For  a  first  example,  let  us  take  the  one  above  stated.  If 
oats  worth  3^.  per  bushel  be  mixed  wi\h.  wheat  worth  9^., 
how  much  must  be  taken  of  each  sort  that  the  compound 
may  be  worth  5^.  per  bushel  ? 


3- 


9- 


4  Oats. 


2  Wheat. 


If  the  price  of  the  mixture  were 
Qs.,  half  the  sum  of  the  prices  of  the 
simples,  it  is  plain  that  it  would  be 
necessary  to  take  just  as  much  oats  as  wheat. 

But  since  the  price  of  the  mixture  is  nearer  to  the  price 
of  the  oats  than  to  that  of  the  wheat,  less  wheat  will  be  re- 
quired in  the  mixture  than  oats. 

Having  set  down  the  prices  of  the  simples  under  each 
other,  and  linked  them  together,  we  next  set  5^.,  the  price 
of  the  mixture,  on  the  left.  We  then  take  the  difference  be- 
tween 9  and  5  and  place  it  opposite  3,  the  price  of  the  oats, 
and  also  the  difference  between  5  and  3,  and  place  it  oppo- 
site 9,  the  price  of  the  wheat.  The  difference  standing  op- 
posite each  kind  shows  how  much  of  that  kind  is  to  be  taken. 
In  the  present  example,  the  mixture  will  consist  of  4  bushels 
of  oats  and  2  of  wheat ;  and  a,ny  other  quantities,  bearing 
the  same  proportion  to  each  other,  such  as  8  and  4,  20  and 
10,  (fee,  will  give  a  mixture  of  the  same  value. 

PROOF    BV    ALLIGATION    MEDIAL. 


4  bushels  of  oats  at  3^. 

-       -       12^. 

2  bushels  of  wheat  at  9^. 

-       -       18^. 

6 

6)30 

Ans,  5^. 

Quest. — 251.  What  is  Alligation  Alternate?     How  do  you  prove  Alli- 
gation Alternate  ? 


ALLIGATION    ALTERNATE.  27S 


252.  To  find  the  proportion  in  which  several  simples  of 
given  prices  must  be  mixed  together,  that  the  compound  ma^ 
be  worth  a  given  price. 

I.  Set  doion  the  prices  of  the  simples  one  under  the  other 
in  the  order  of  their  values^  beginning  with  the  /Lowest, 

II.  Linh  the  least  price  with  the  greatest,  and  the  one  nex. 
to  the  least  with  the  one  next  to  the  greatest,  and  so  on,  until 
the  price  of  each  simple  which  is  less  than  the  price  of  th 
mixture  is  linked  with  one  or  more  thai  is  greater;  and  everi 
one  that  is  greater  with  one  or  more  that  is  less, 

III.  Write  the  difference  between  the  price  of  the  mixture  ano 
that  of  each  of  the  simples  opposite  that  price  with  which  tht 
particular  simple  is  linked ;  then  the  difference  standing  opposite 
any  one  price,  or  the  sum  of  the  differences  when  there  is  mon 
than  one,  will  express  the  quantity  to  be  taken  of  that  price, 

EXAMPLES. 

1.  A  merchant  would  mix  wines  worth  16^.,  18^.,  and  22^. 
per  gallon  in  such  a  way,  that  the  mixture  may  be  worth  20s 
per  gallon :  how  much  must  be  taken  of  each  sort  ? 


16- 
20^  18-1 
22^ 


2  at  16^. 
2  at  18^. 
4  +  2  =  6  at  225. 
.         ^  2gal.  at  16^.,  2  at  18^.,  and  6  at  22^. :  or  any  other 
\       quantities  bearing  the  proportion  of  2,  2,  and  6. 

2.  What  proportions  of  coffee  at  Sets.,  \Octs.,  and  lActs, 
per  lb.  must  be  mixed  together  so  that  the  compound  shall  be 
worth  I2cts.  per  lb.1 

3.  A  goldsmith  has  gold  of  16,  of  18,  of  23,  and  of  24 
carats  fine :  what  part  must  be  taken  of  each  so  that  the 
mixture  shall  be  21  carats  fine  ? 

Quest. — 252.  How  do  you  find  the  proportions  so  that  the  compound  may 
be  of  a  given  price  ? 


280 


ALLIGATION    ALTERNATE. 


4.  What  portion  of  brandy  at  14^.  per  gallon,  of  old  Ma- 
deira at  24s.  per  gallon,  of  new  Madeira  at  21^.  per  gallon, 
and  of  brandy  at  10^.  per  gallon,  must  be  mixed  together  so 
that  the  mixture  shall  be  worth  1 8^.  per  gallon  1 


253.  When  a  given  quantity  of  one  of  the  simples  is  to  be 
taken. 

I.  Find  the  proportional  quantities  of  the  simples  as  in 
Case  I, 

II.  Then  saj/,  as  th^  number  opposite  the  simple  whose  quan- 
tity is  given,  is  to  either  proportional  quantity,  so  is  the  given 
quantity,  to  the  proportional  part  of  the  corresponding  simple, 

EXAMPLES. 

1.  How  much  wine  at  5s.,  at  bs,  6d.,  and  6s,  per  gallon 
must  be  mixed  with  4  gallons  at  4^.  per  gallon,  so  that  the 
mixture  shall  be  worth  5^.  4c?.  per  gallon  ? 

48- 


64^ 


-  simple  whose  quantity  is  known, 
proportional  quantities. 


2 

4 
16 


Ans.  Igal.  at  5^.,  2  at  5^.  6.,  and  8  at  6s 


PROOF    BY    ALLIGATION    MEDIAL. 

4gaL  at  4^.  per  gallon    -  -  I92d. 

1  "      5s.  " 60 

2  "      5s.  6d.       "     -  -  -  132 
8      "      6s.  «     -  -  .  576 

IF 


15)960(64^.  price  of  mixture. 


Quest. — ^253.  How  do  you  find  the  proportion  when  the  quantity  of  one 
of  the  simples  is  given  ? 


ALLIGATION    ALTERNATE. 


281 


2.  A  farmer  would  mix  14  bushels  of  wheat  at  $1,20  per 
bushel,  with  rye  at  12cts.,  barley  at  AQcts.,  and  oats  at  ZQcts, : 
how  much  must  be  taken  of  each  sort  to  make  the  mixture 
worth  64  cents  per  bushel  ? 

3.  There  is  a  mixture  made  of  wheat  at  4^.  per  bushel, 
rye  at  3^.,  barley  at  2^.,  with  12  bushels  of  oats  at  \Qd.  per 
bushel :  how  much  has  been  taken  of  each  sort  when  the 
mixture  is  worth  3^.  6J.  ? 

4.  A  distiller  would  mix  AOgah  of  French  brandy  at  12^. 
per  gallon,  with  English  at  7^.  and  spirits  at  4^.  per  gallon : 
what  quantity  must  be  taken  of  each  sort,  that  the  mixture 
may  be  afforded  at  8^.  per  gallon  ? 


254.  When  the  quantity  of  the  compound  is  given  as  web 
as  the  price. 

I.  Find  the  proportional  quantities  as  in  Case  I, 
,     II.   Then  say,  as  the  sum  of  the  proportional  quantities^  is 
to  each  proportional  quantity^  so  is  the  given  quantity,  to  the 
corresponding  part  of  each, 

EXAMPLES. 


1.  A  grocer  has  four  sorts  of  sugar  worth  12d.j  lOd.,  6d., 
and  Ad.  per  pound  ;  he  would  make  a  mixture  of  144Z5.  worth 
Sd.  per  pound :  what  quantity  must  be  taken  of  each  sort  ? 


8. 


^1 

Ll2 

Sum  of  the  proportional  ^aiia 

Ans, 


12 
12 
12 
12 


4 
2 
2 

_4 

12 

(48Z5.    at    Ad. 

I  2Alh,  at  lOd, ; 


48 
24 
24 
48 


;     2Alh,    at    6^/. ; 
and  ASlb,  at  \2d. 


:     A     : 

144     : 

:     2     : 

144     : 

:     2     : 

144     : 

:     4     : 

144     : 

Quest. — ^254.  How  do  you  determine  the  proportion  when  the  quantity 
of  the  compound  is  given  as  well  as  the  price  ? 


282  CUSTOM    HOUSE    BUSINESS. 

PROOF    BY    ALLIGATION    MEDIAL. 

48ZJ.  at     4fZ.  -  -  -  -  I92d. 

24lb.  "      6^.  -  -  .  -  I44d. 

24Z5.  "    lOd.  -  -  -  -  240d. 

48lb.  "    I2d,  -  -  .  -  57ed. 


144  144)1 152(8cZ. 

Hence,  the  average  cost  is  8d. 

2.  A  grocer  having  four  sorts  of  tea  worth  5s.,  6s.,  8s.y 
and  9^.  per  lb.,  wishes  a  mixture  of  87/^.  worth  7^.  per  lb, : 
how  much  must  be  taken  of  each  sort  ? 

3.  A  vintner  has  four  sorts  of  wine,  viz.,  white  wine  at  4^, 
per  gallon,  Flemish  at  6s.  per  gallon,  Malaga  at  8^.  per  gal- 
lon, and  Canary  at  10^.  per  gallon  :  he  would  make  a  mixture 
of  60  gallons  to  be  worth  5^.  per  gallon  :  what  quantity  must 
be  taken  of  each  ? 

4.  A  silversmith  has  four  sorts  of  gold,  viz.,  of  24  carats 
fine,  of  22  carats  fine,  of  20  carats  fine,  and  of  15  carats  fine  : 
he  would  make  a  mixture  of  42oz,  of  17  carats  fine  :  how 
much  must  be  taken  of  each  sort  ? 


CUSTOM  HOUSE  BUSINESS. 

255.  Persons  who  bring  goods,  or  merchandise,  into  the 
United  States,  from  foreign  countries,  are  required  to  land 
them  at  particular  places  or  ports,  called  Ports  of  Entry,  and 
to  pay  a  certain  amount  on  their  value,  called  a  Duty.  This 
duty  is  imposed  by  the  General  Government,  and  must  be  the 
same  on  the  same  articles  of  merchandise,  in  every  part  of 
the  United  States. 

Besides  the  duties  on  merchandise,  vessels  employed  in 
commerce  are  required,  by  law,  to  pay  certain  sums  for  the 
privilege  of  entering  the  ports.     These  sums  are  large  or 

Quest. — 255.  What  is  a  port  of  entry?  What  is  a  duty?  By  whom  are 
duties  imposed  ?  What  charges  are  vessels  required  to  pay  ?  What  are 
the  moneys  arising  from  duties  ind  tonnage  called  ? 


CUSTOM    HOUSE    BUSINESS.  288 

small,  in  proportion  to  the  size  or  tonnage  of  vessels.  The 
moneys  arising  from  duties  and  tonnage,  are  called  revenues, 
256.  The  revenues  of  the  country  are  under  the  general, 
direction  of  the  Secretary  of  the  Treasury,  and  to  secure 
their  faithful  collection,  the  government  has  appointed  va- 
rious officers  at  each  port  of  entry  or  place  where  goods  may 
be  landed. 

1  257.  The  office  established  by  the  government  at  any  port 
of  entry,  is  called  a  Custom  House,  and  the  officers  attached 
to  it  are  called  Custom  House  Officers. 

258.  All  duties  levied  by  law  on  goods  imported  into  the 
United  States,  are  collected  at  the  various  custom  houses, 
and  are  of  two  kinds,  Specific  and  Ad  valorem. 

A  specific  duty  is  a  certain  sum  on  a  particular  kind  of 
goods  named  ;  as  so  much  per  square  yard  on  cotton  or 
woollen  cloths,  so  much  per  ton  weight  on  iron,  or  so  much 
per  gallon  on  molasses. 

An  ad  valorem  duty  is  such  a  per  cent  on  the  actual  cost 
of  the  goods  in  the  country  from  which  they  are  imported. 
Thus,  an  ad  valorem  duty  of  15  per  cent  on  English  cloths, 
is  a  duty  of  15  per  cent  on  the  cost  of  cloths  imported  from 
England. 

259.  The  laws  of  Congress  provide,  that  the  cargoes  of 
all  vessels  freighted  with  foreign  goods  or  merchandise,  shall 
be  weighed  or  gauged  by  the  custom  house  officers  at  the 
port  to  which  they  are  consigned.  As  duties  are  only  to  be 
paid  on  the  articles,  and  not  on  the  boxes,  casks,  and  bags 
which  contain  them,  certain  deductions  are  made  from  the 
weights  and  measures,  called  Allowances. 

Gross  Weight  is  the  whole  weight  of  the  goods,  together 

Quest. — 256.  Under  whose  direction  are  the  revenues  of  the  country  ? 

257.  What  is  a  custom  house?    What  are  the  officers  attached  to  it  called? 

258.  Where  are  the  duties  collected?  How  many  kinds  are  there,  and 
what  are  they  called  ?     What  is  a  specific  duty  ?     An  ad  valorem  duty  ? 

259.  What  do  the  laws  of  Congress  du*ect  in  relation  to  foreign  goods  ?  Why 
are  deductions  made  from  their  weight?  What  are  these  deductions 
called?     What  is  gross  weight? 


284  CUSTOM    HOUSE    BUSINESS. 

with  that  of  the  hogshead,  barrel,  box,  bag,  &c.,  which  con- 
tains them. 

Draft  is  an  alloAvance  from  the  gross  weight  on  account 
of  waste,  where  there  is  not  actual  tare. 

Ih.  lb. 

On        112  it  is    1, 

From      112  to     224       "      2, 

224  to     336       "       3, 

"  336  to  1120       "       4, 

"        1120  to  2016       "      7, 

Above  2016  any  weight"      9; 

consequently,  9/^.  is  the  greatest  draft  allowed. 

Tare  is  an  allowance  made  for  the  weight  of  the  boxes,  bar- 
rels, or  bags  containing  the  commodity,  and  is  of  three  kinds. 
1st.  Legal  tare,  or  such  as  is  established  by  law;  2d.  Cus- 
tomary tare,  or  such  as  is  established  by  the  custom  among 
mercliants  ;  and  3d.  Actual  tare,  or  such  as  is  found  by  re- 
moving the  goods  and  actually  weighing  the  boxes  or  casks 
m  which  they  are  contained. 

On  liquors  in  casks,  customary  tare  is  sometimes  allowed 
on  the  supposition  that  the  cask  is  not  full,  or  what  is  called 
its  actual  wants;  and  then  an  allowance  of  5  per  cent  for 
leakage. 

A  tare  of  10  per  cent  is  allowed  on  porter,  ale,  and  beer, 
in  bottles,  on  account  of  breakage,  and  5  per  cent  on  all 
other  liquors  in  bottles.  At  the  custom  house,  bottles  of  the 
common  size  are  estimated  to  contain  2|-  gallons  the  dozen. 
For  tables  of  Tare  and  Duty,  see  Ogden  on  theTarifF  of  1842. 

EXAMPLES. 

1.  What  will  be  the  duty  on  125  cartons  of  ribbons,  each 
containing  48  pieces,  and  each  piece  weighing  3oz,  net,  and 
paying  a  duty  of  $2,50  per  lb.  ? 

Quest. — What  is  draft  ?  What  is  the  greatest  draft  allowed  ?  What  is 
tare?  What  are  the  different  kuids  of  tare?  What  allowances  are  made 
on  liquors  ? 


FORMS    RELATING    TO    BUSINESS    IN    GENERAL.       285 

2.  What  will  be  the  duty  on  225  bags  of  coffee,  each 
weighing  gross  160Z^.,  invoiced  at  6  cents  per  lb,;  2  per 
cent  being  the  legal  rate  of  tare,  and  20  per  cent  the  duty  ? 

3.  What  duty  must  be  paid  on  275  dozen  bottles  of  claret, 
estimated  to  contain  2^  gallons  per  dozen,  5  per  cent  being 
allowed  for  breakage,  and  the  duty  being  35  cents  per  gallon  ? 

4.  A  merchant  imports  175  cases  of  indigo,  each  case 
weighing  I96lb.  gross :  15  per  cent  is  the  customary  rate  of 
tare,  and  the  duty  5  cents  per  lb.  What  duty  must  he  pay  on 
the  whole  ? 

5.  What  is  the  tare  and  duty  on  75  casks  of  Epsom  salts, 
each  weighing  gross  2cwt.  2qr.  27lb.,  and  invoiced  at  1|- 
cents  per  lb.,  the  customary  tare  being  11  per  cent,  and  the 
rate  of  duty  20  per  cent  1 


FORMS  RELATING  TO  BUSINESS  IN  GENERAL. 


FORMS    OF    ORDERS  - 

Messrs.  M.  James  &  Co. 

Please  pay  John  Thompson,  or  order,  five  hundred 
dollars,  and  place  the  same  to  my  account,  for  value  received. 

Peter  Worthy. 
Wilmington,  N.  C,  June  1,  1846. 


Mr.  Joseph  Rich, 

Please  pay,  for  value  received,  the  bearer,  sixty-one 
dollars  and  twenty  cents,  in  goods  from  your  store,  and  charge  the 
same  to  the  account  of  your  Obedient  Servant, 

John  Parsons. 
Savannah,  Ga.,  July  1,  1846. 


FORMS    OF    RECEIPTS. 

Receipt  for  Money  on  Account. 

Received,  Natchez,  June  2d,  1845,  of  John  Ward,  sixty  dollars 
on  account. 


$60,00  John  P.  Fay. 


286       FORMS   RELATING    TO    BUSINESS    IN    GENERAL. 

Receipt  for  Money  on  a  Note. 
Received,  Nashville,  June  5,  1846,  of  Leonard  Walsh,  six  hun- 
dred and  forty  dollars,  on  his  note  for  one  thousand  dollars,  date^ 
New  York,  January  1,  1845. 

$640,00  J.  N.  Weeks. 


A    BOND    FOR    ONE    PERSON,    WITH    A    CONDITION. 

KNOW  ALL  MEN  BY  THESE  PRESENTS,  That,  1 
James  Wilson  of  the  City  of  Hartford  and  State  of  Connecticut , 
am  held  and  firmly  bound  unto  John  Pickens  of  the  Town  of  Water- 
hury,  County  of  New  Haven  and  State  of  Connecticut^  in  the  sum 
of  Eighty  dollars  lawful  money  of  the  United  States  of  America,  to 
be  paid  to  the  said  John  Pickens,  his  executors,  administrators,  or 
assigns  :  for  which  payment  well  and  truly  to  be  made  /  bind  my- 
self  my  heirs,  executors,  and  administrators,  firmly  by  these  pres- 
ents. Sealed  with  my  Seal.  Dated  the  Ninth  day  of  March  one 
thousand  eight  hundred  and  thirty-eight. 

THE  CONDITION  of  the  above  obligation  is  such,  that  if  the 
above  bounden  James  Wilson,  his  heirs,  executors,  or  administra- 
tors, shall  well  and  truly  pay  or  cause  to  be  paid,  unto  the  above 
named  John  Pickens,  his  executors,  administrators,  or  assigns,  the 
just  and  full  sum  of 

Here  insert  the  condition. 


then  the  above  obligation  to  be  void,  otherwise  to  remain  in  full 

force  and  virtue. 

Sealed  and  delivered  in 

the  presence  of  .    .     .^ 

John  Frost,  >  James  Wilson,  ^t^i 

Joseph  Wiggins,  S  SBn^ 

Note. — The  part  in  Italic  to  be  filled  up  according  to  circum- 
stance. 

If  there  is  no  condition  to  the  bond,  then  all  to  be  omitted  after 
and  including  the  words  "  THE  CONDITION,  &c." 


FORMS   RELATING    TO    BUSINESS    IN    GENERAL.       287 


A    BOND    FOR    TWO    PERSONS,   WITH    A    CONDITION. 

KNOW  ALL  MEN  BY  THESE  PRESENTS,  That,  We 
Tames  Wilson  and  Thomas  Ash  of  the  City  of  Hartford  and  State 
of  Connecticut,  are  held  and  firmly  bound  unto  John  Pickens  of  the 
Town  of  Waterhury  County  of  New  Haven  and  State  of  Connecti- 
cut, in  the  sum  of  Eighty  dollars  lawful  money  of  the  United  States 
of  America,  to  be  paid  to  the  said  John  Pickens,  his  executors  or 
assigns :  for  which  payment  well  and  truly  to  be  made  We  bind 
ourselves,  our  heirs,  executors,  and  administrators,  firmly  by  these 
presents.  Sealed  with  our  Seal.  Dated  the  Ninth  day  of  March 
one  thousand  eight  hundred  and  thirty-eight, 

THE  CONDITION  of  the  above  obligation  is  such,  that  if  the 
above  bounden  James  Wilson  and  Thomas  Ash,  their  heirs,  execu- 
tors, or  administrators,  shall  well  and  truly  pay  or  cause  to  be  paid, 
unto  the  above  named  John  Pickens,  his  executors,  administrators, 
or  assigns,  the  just  and  full  sum  of 


Here  insert  the  condition. 


then  the  above  obligation  to  be  void,  otherwise  to  remain  in  full 
force  and  virtue. 

Sealed  and  delivered  in 
the  presence  of 

John  Frost,  >  James  Wilson,         ^Hlvifc 

Joseph  Wiggins,  J  Thomas  Ash.  MWg^J^ 

Note. — The  part  in  Italic  to  be  filled  up  according  to  circum- 
stance. 

If  there  is  no  condition  to  the  bond,  then  all  to  be  omitted  after 
and  including  the  words  "  THE  CONDITION,  &c." 


288  GENERAL   AVERAGE 


GENERAL  AVERAGE. 


260.  Average  is  a  term  of  commerce  and  navigation,  to 
signify  a  contribution  by  individuals,  where  the  goods  of  a 
particular  merchant  are  thrown  overboard  in  a  storm,  to  save 
the  ship  from  sinking,  or  where  the  masts,  cables,  anchors, 
or  other  furniture  of  the  ship  are  cut  away  or  destroyed  for 
the  preservation  of  the  whole.  In  these  and  like  cases, 
where  any  sacrifices  are  deliberately  made,  or  any  expenses 
voluntarily  incurred,  to  prevent  a  total  loss,  such  sacrifice  or 
expense  is  the  proper  subject  of  a  general  contribution,  and 
ought  to  be  rateably  borne  by  the  owners  of  the  ship,  the 
freight,  and  the  cargo,  so  that  the  loss  may  fall  proportion- 
ably  on  all.    The  amount  sacrificed  is  called  the  jettison. 

261.  Average  is  either  general  or  particular ;  that  is,  it  is 
either  chargeable  to  all  the  interests,  viz.,  the  ship,  the 
freight,  and  the  cargo,  or  only  to  some  of  them.  As  when 
losses  occur  from  ordinary  wear  and  tear,  or  from  the  perils 
incident  to  the  voyage,  without  being  voluntarily/  incurred ; 
or  when  any  particular  sacrifice  is  made  for  the  sake  of  the 
ship  only  or  the  cargo  only,  these  losses  must  be  borne  by  the 
parties  immediately  interested,  and  are  consequently  defrayed 
by  a  particular  average.  There  are  also  some  small  charges 
called  petty  or  accustomed  averages,  one-third  of  which  is 
usually  charged  to  the  ship  and  two-thirds  to  the  cargo. 

No  general  average  ever  takes  place,  except  it  can  be 
shown  that  the  danger  was  imminent,  and  that  the  sacrifice  was 
made  indispensable,  or  supposed  to  be  so  by  the  captain  and  offi- 
cers, for  the  safety  of  the  ship. 

262.  In  different  countries  different  modes  are  adopted  of 
valuing  the  articles  which  are  to  constitute  a  general  aver- 
age. In  general,  however,  the  value  of  the  freight  is  held  to 
be  the  clear  sum  which  the  ship  has  earned  after  seamen's 

Quest. — 260.  What  does  the  term  average  signify  ?  261.  How  many 
kinds  of  average  are  there  ?  What  are  the  small  charges  called  1  Under 
what  ch-cumstances  will  a  general  average  take  place  ?  262.  How  is  the 
fi^eight  valued?    How  much  is  charged  on  account  of  the  seamen's  wages? 


GENERAL    AVERAGE.  289 

vages,  pilotage,  and  all  such  other  charges  as  came  under 
the  name  of  petty  charges,  are  deducted  ;  one-third,  and  in 
some  cases  one-half,  being  deducted  for  the  wages  of  the  crew. 
The  goods  lost,  as  well  as  those  saved;  are  valued  at  the 
price  they  would  have  brought  in  ready  mone)r  at  the  place 
of  delivery,  on  the  ship's  arriving  there,  freight,  duties,  and 
all  other  charges  being  deducted :  indeed,  they  bear  their 
proportions,  the  same  as  the  goods  saved.  The  ship  is 
♦valued  at  the  price  she  would  bring  on  her  arrival  at  the  port 
of  delivery.  But  when  the  loss  of  masts,  cables,  and  other 
furniture  of  the  ship  is  compensated  by  general  average,  it 
is  usual,  as  the  new  articles  will  be  of  greater  value  than  the 
old,  to  deduct  one-third,  leaving  two  thirds  only  to  be  charged 
to  the  amount  to  be  contributed. 

EXAMPLES. 

1.  The  vessel  Good  Intent,  bound  from  New  York  to  New 
Orleans,  was  lost  on  the  Jersey  beach  the  day  after  sailing. 
She  cut  av.'ay  her  cables  and  masts,  and  cast  overboard  a 
part  of  her  cargo,  by  which  another  part  was  injured.  The 
ship  was  finally  got  off,  and  brought  back  to  New  York. 

AMOUNT    OY    LOSS. 

Goods  of  A  cast  overboard  -  -  -  -  $500 
Damage  of  the  goods  of  B  by  the  jettison  -  200 
Freight  of  the  goods  cast  overboard    -       -r         100 


Cable,  anchors,  mast,  &c.,  worth      ^^^^^^  r  „^^ 
Deduct  one-third      -       -       -       -        100  i 

Expenses  of  getting  the  ship  off  the  sands  56 

Pilotage  and  port  duties  going  in  and  out  >  .  ^^ 
of  the  harbor,  commissions,  &c.      -        S 

Expenses  in  port 25 

Adjusting  the  average     -----  4 

Postage 1 

Total  loss  $1186 

Quest. — How  is  the  cargo  valued  ?     Does  the  part  lost  bear  its  part  of 
h\Q  loss  1     How  is  the  ship  valued  ?     When  parts  of  the  ship  are  lost,  how 
are  they  compensated  for?     How  do  you  explain  the  example? 
13 


290  GENERAL    AVERAGE. 

ARTICLES    TO    CONTRIBUTE. 

Goods  of  A  cast  overboard     - 

Value  of  B's  goods  at  N.  O.,  deducting  freight,  &c.  1000 

'*      ofC's            '*              "               "               "  500 

u      ofD»s           u             .i               u               u  2000 

"      ofE's           "             "               "               "  5000 

Value  of  the  ship 2000 

Freight  after  deducting  one-third 800 

$11,800 
Then,  total  value  :  total  loss  :  :  100  :  per  cent  of  loss. 
$11800  :  1180  :  :  100  :  10; 
hence,  each  loses  10  per  cent  on  the  value  of  his  interest  m 
the  cargo,  ship,  or  freight.  Therefore,  A  loses  $50,  B  $100, 
C  $50,  D  $200,  E  $500,  the  owners  of  the  ship  $280— in 
all  $1180.  Upon  this  calculation  the  owners  are  to  lose 
$280 ;  but  they  are  to  receive  their  disbursements  from  the 
contribution,  viz.,  freight  on  goods  thrown  overboard  $100, 
damages  to  ship  $200,  various  disbursements  in  expenses 
$180,  total  $480  ;  and  deducting  the  amount  of  contribution, 
they  will  actually  receive  $200.  Hence,  the  account  will 
stand : 

The  owners  are  to  receive $200 

A  loses  $500,  and  is  to  contribute     $50  ;   hence,  )         . -^ 

he  receives- 3 

B  loses  $200,  and  is  to  contribute  $100  ;  hence,  >         ,^^ 

he  receives S 

Total  to  be  received       -       -       -     $750 

^   C  $  50 
C,  D,  and  E  have  lost  nothing,  and  are  to  pay      <   D     200 

(  E     500 

Total  actually  paid     -       -       -        $750 ; 

so  that  the  total  to  be  paid  is  just  equal  to  the  total  loss,  as  it 
should  be,  and  A  and  B  get  their  remaining  and  injured  goods, 
and  the  three  others  get  theirs  in  a  perfect  state,  after  paying 
their  rateable  proportion  of  the  loss. 


TONNAGE    OF    VESSELS.  291 


TONNAGE  OF  VESSELS. 

263.  There  are  certain  custom  house  charges  on  vessels, 
which  are  made  according  to  their  tonnage.  The  tonnage 
of  a  vessel  is  the  number  of  tons  weight  she  will  carry,  and 
this  is  determined  by  measurement. 

[From  the  "  Digest,"  by  Andrew  A.  Jones,  Esq.,  of  the  N.  Y.  Custom  House.] 

Custom  house  charges  on  all  ships  or  vessels  entering  from  any 
foreign  port  or  place. 

Skips  or  vessels  of  the  United  States,  having  three-fourths 

of  the  crew  and  all  the  officers  American  citizens,  per  ton  $0,06 
Ships  or  vessels  of  nations  entitled  by  treaty  to  enter  at  the 

same  rate  as  American  vessels       -         -         -         -         -        ,06 
Ships  or  vessels  of  the  United  States  not  having  three- 
fourths  the  crew  as  above ,50 

On  foreign  ships  or  vessels  other  than  those  entitled  by 

treaty  -------.--,50 

Additional  tonnage  on  foreign  vessels,  denominated  light 

money  --------_, 50 

Licensed  coasters  are  also  liable  once  in  each  year  to  a  duty  of 
50  cents  per  ton,  being  engaged  in  a  trade  from  a  port  in  one  state 
to  a  port  in  another  state,  other  than  an  adjoining  state,  unless  the 
officers  and  three- fourths  of  the  crew  are  American  citizens ;  to 
ascertain  which,  the  crews  are  always  liable  to  an  examination  by 
an  officer. 

A  foreign  vessel  is  not  permitted  to  carry  on  the  coasting  trade  ; 
but  having  arrived  from  a  foreign  port  with  a  cargo  consigned  to 
more  than  one  port  of  the  United  States,  she  may  proceed  coast- 
wise with  a  certified  manifest  until  her  voyage  is  completed. 

264.  The  government  estimate  the  tonnage  according  to 
one  rule,  while  the  ship  carpenter  who  builds  the  vessel 
uses  another. 

Quest. — 263.  What  is  the  tonnage  of  a  vessel  ?  What  are  the  custom 
house  charges  on  the  different  classes  of  vessels  trading  with  foreign  coun 
tries?     To  what  charores  arc  coasters  subject? 


292  TONNxVGE    OF    VESSELS. 

Government  Rule.  I.  Measure,  in  feet,  above  the  upper 
deck  the  length  of  the  vessel,  from  the  fore  part  of  the  main  stem 
to  the  after  part  of  the  stern  post.  Then  measure  the  breadth 
taken  at  the  widest  part  above  the  main  wale  on  the  outside,  and 
the  depth  from  the  under  side  of  the  deck  plank  to  the  ceiling  in 
the  hold. 

II.  From  the  length  take  three-ffths  of  the  breadth  and  mul- 
tiply the  remainder  by  the  breadth  and  depth,  and  the  product 
divided  by  95  will  give  the  tonnage  of  a  single  decker ;  and  the 
same  for  a  double  decker,  by  merely  making  the  depth  equal  to 
half  the  breadth. 

Carpenters'  Rule.  Multiply  together  the  length  of  the 
keel,  the  breadth  of  the  main  beam,  and  the  depth  of  the  hold, 
and  the  product  divided  by  95  will  be  the  carpenters^  tonnage  foi 
a  single  decker ;  and  for  a  double  decker,  deduct  from  the  depth 
of  the  hold  half  the  distance  betimen  decks. 

EXAillPLES. 

1.  What  is  the  government  tonnage  of  a  single  decker, 
whose  length  is  75  feet,  breadth  20  feet,  and  depth  17  feet? 

2.  What  is  the  carpenters'  tonnage  of  a  single  decker,  the 
length  of  whose  keel  is  90  feet,  breadth  22  feet  7  inches,  and 
depth  20  feet  6  inches  ? 

3.  What  is  the  carpenters'  tonnage  of  a  steamship,  double 
decker,  length  154  feet,  breadth  30  feet  8  inches,  and  depth 
after  deducting  half  between  decks,  14  feet  8  inches? 

4.  What  is  the  government  tonnage  of  a  double  decker, 
the  length  being  103  feet,  breadth  25  feet  6  inches  ? 

5.  What  is  the  carpenters'  tonnage  of  a  double  decker,  its 
length  125  feet,  breadth  25  feet  6  inches,  entire  depth  34 
feet,  and  distance  between  decks  8  feet  ? 

Quest. — 2f)4.  What  is  the  government  rule  for  fmdmg  the  tonnage? 
What  the  ship-builders'  rule. 


GAUGING 


293 


GAUGING. 

265.  Cask-gauging  is  the  method  of  finding  the  number 
of  gallons  which  a  cask  contains,  by  measuring  the  external 
dimensions  of  the  cask. 

266.  Casks  are  divided  into  four  varieties,  according  to  the 
curvature  of  their  sides.  To  which  of  the  varieties  any  cask 
belongs,  must  be  judged  of  by  inspection. 


1.  Of  the  least  curvature. 


2d  Variety. 


3d  Variety. 


4th  Variety. 


267.  The  first  thing  to  be  done  is  to  find  the  mean  diame- 
ter.    To  do  this. 

Divide  the  head  diameter  hy  the  hung  diameter^  and  find  the 
quotient  in  the  first  column  of  the  following  table,  marked  Qu. 
Then  if  the  hung  diameter  he  multiplied  hy  the  number  on  the 
same  line  with  it,  and  in  the  column  answering  to  the  proper 

Quest. — 265.  What  is  cask-gau^ng  ?  266.  Into  how  many  varieties 
are  casks  divided  ?     267    How  do  you  find  the  mean  diameter  ? 


294 


GAUGING. 


variety f  the  product  will  he  the  true  mean  diameter,  or  the  diam^ 
eter  of  a  cylinder  of  the  same  content  with  the  cask  proposed, 
cutting  off  four  figures  for  decimals. 


au. 

IstVar. 

2(1  Var. 

3d  Var. 

4th  Var. 

au. 

1st  Var. 

2d  Var. 

3d  Var. 

4th  Var. 

60 

8660 

8465 

7905 

7637 

76 

9270 

9227 

8881 

8827 

51 

8680 

8493 

7937 

7681 

77 

9296 

9258 

8944 

8874 

52 

8700 

8520 

7970 

7725 

78 

9324 

9290 

8967 

8922 

53 

8720 

8548 

8002 

7769 

79 

9352 

9320 

9011 

8970 

54 

8740 

8576 

8036 

7813 

80 

9380 

9352 

9055 

9018 

55 

8760 

8605 

8070 

7858 

81 

9409 

9383 

9100 

9066 

56 

8781 

8633 

8104 

7902 

82 

9438 

9415 

9144 

9114 

57 

8802 

8662 

8140 

7947 

83 

9467 

9446 

9189 

9163 

58 

8824 

8690 

8174 

7992 

84 

9496 

9478 

9234 

9211 

59 

8846 

8720 

8210 

8037 

85 

9526 

9510 

9280 

9260 

60 

8869 

8748 

8246 

8082 

86 

9556 

9542 

9326 

9308 

61 

8892 

8777 

8282 

8128 

87 

9586 

9574 

9372 

9357 

62 

8915 

8806 

8320 

8173 

88 

9616 

9606 

9419 

9406 

63 

8938 

8835 

8357 

8220 

89 

9647 

9638 

9466 

9455 

64 

8962 

8865 

8395 

8265 

90 

9678 

9671 

9513 

9504 

65 

8986 

8894 

8433 

8311 

91 

9710 

9703 

9560 

9553 

66 

9010 

8924 

8472 

8357 

92 

9740 

9736 

9608 

9602 

67 

9034 

8954 

8511 

8404 

93 

9772 

9768 

9656 

9652 

68 

9060 

8983 

8551 

8450 

94 

9804 

9801 

9704 

9701 

69 

9084 

9013 

8590 

8497 

95 

9836 

9834 

9753 

9751 

70 

9110 

9044 

8631 

8544 

96 

9868 

9867 

9802 

9800 

71 

9136 

9074 

8672 

8590 

97 

9901 

9900 

9851 

9850 

72 

9162 

9104 

8713 

8637 

98 

9933 

9933 

9900 

9900 

73 

9188 

9135 

8754 

8685 

99 

9966 

9966 

9950 

9950 

74 

9215 

9166 

8796 

8732 

100 

10000 

10000 

10000 

10000 

75 

9242 

9196 

8838 

8780 

EXAMPLES. 


1 .  Supposing  the  diameters  to  be  32  and  24,  it  is  required 
to  find  the  mean  diameter  for  each  variety. 

Dividing  24  by  32,  we  obtain  .75  ;  which  being  found  in 
the  column  of  quotients,  opposite  thereto  stand  the  numbers, 


9242  1      ,.  ^  ^  .  ^       ,     r  39.5744 

9196       ^^!"^  being  each  mul-       39  4373 
««QQ    M'Pl'^d    by   32,   produce  ^_    9,Q9,Rlfi 


.8838  r-^---    "^ 
L. 8780  J  ^^^P^^^^^^^y^ 


28.2816 
L  28.0960 


for  the  correspond- 
■  ing   mean    diame- 
ters required. 


OPERATION. 

I  X  (f  X  .0034. 


GAUGING.  295 

2.  The  head  diameter  of  a  cask  is  26  inches,  and  the 
bung  diameter  3  feet  2  inches :  what  is  the  mean  diameter, 
the  cask  being  of  the  third  variety  1 

3.  The  head  diameter  is  22  inches,  the  bung  diameter  34 
inches  :  what  is  the  mean  diameter  of  a  cask  of  the  fourth 
variety  1 

268.  Having  found  the  mean  diameter,  we  multiply  the 
square  of  the  mean  diameter  by  the  decimal  .7854,  and  the 
product  by  the  length ;  this  will  give  the  solid  content  in  cu- 
bic inches.  Then  if  we  divide  by  231,  we  have  the  content 
in  wine  gallons  (see  Art.  31),  or  if  we  divide  by  282,  we  have 
the  content  in  beer  gallons. 

For  wine  measure  we  multiply 
the  length  by  the  square  of  the 
mean  diameter,  then  by  the  deci- 
mal .7854,  and  divide  by  231. 

If,  then,  we  divide  the  decimal  .7854  by  231,  the  quotient 
carried  to  four  places  of  decimals  is  .0034,  and  this  decimal 
multiplied  by  the  square  of  the  mean  diameter  and  by  the 
length  of  the  cask,  will  give  the  content  in  wine  gallons. 

For  similar  reasons,  the  con- 
tent is  found  in  beer  gallons  by 
multiplying  together  the  length, 
the  square  of  the  mean  diameter, 
and  the  decimal  .0028. 

Hence,  for  gauging  or  measuring  casks, 

Multiply  the  length  hy  the  square  of  the  mean  diameter ;  then 
multiply  hy  Z^  for  wifie,  and  hy  28  for  heer  measure^  and  point 
off  in  the  product  four  decimal  places.  The  product  will  then 
express  gallons  and  the  decimals  of  a  gallon. 

-.  How  many  wine  gallons  in  a  cask,  whose  bung  diame- 
ter is  36  inches,  head  diameter  30  inches,  and  length  50 
inches  ;  the  cask  being  of  the  first  variety  ? 

Quest. — ^268.  How  do  you  find  the  solidity  1  How  do  you  find  the  con- 
tent in  wine  gallons  1     In  beer  gallons  1 


OPERATION. 
/  X  ^  X  -^2^*  = 
IX  d}  X  .0028. 


296  LIFE    INSURANCE. 

2.  What  is  the  number  of  beer  gallons  in  the  last  example  ? 

3.  How  many  wine,  and  how  many  beer  gallons  in  a  cask 
whose  length  is  36  inches,  bung  diameter  35  inches,  and  head 
diameter  30  inches,  it  being  of  the  first  variety? 

4.  How  many  wine  gallons  in  a  cask  of  which  the  head 
diameter  is  24  inches,  bung  diameter  36  inches,  and  length 
3  feet  6  inches,  the  cask  being  of  the  second  variety  ? 


LIFE  INSURANCE. 

269.  Insurance  for  a  term  of  years,  or  for  the  entire  con- 
tinuance of  life,  is  a  contract  on  the  part  of  an  authorized 
association  to  pay  a  certain  sum,  specified  in  the  policy  of 
insurance,  on  the  happening  of  an  event  named  therein,  and 
for  which  the  association  receives  a  certain  premium,  gen- 
erally in  the  form  of  an  annual  payment. 

270.  To  enable  the  company  to  fix  their  premiums  at  such 
rates  as  shall  be  both  fair  to  the  insured  and  safe  to  the  asso- 
ciation, they  must  know  the  average  duration  of  life  from  its 
commencement  to  its  extreme  limit.  This  average  is  called 
the  "  Expectation  of  Life^''  and  this  is  determined  by  col- 
lecting from  many  sources  the  most  authentic  information  in 
regard  to  births  and  deaths.  The  "  Carlisle  Table,"  which  is 
subjoined,  and  which  shows  the  expectation  of  life  from  birth 
to  103  years,  is  considered  the  most  accurate.  It  is  much 
used  in  England,  and  is  in  general  use  in  this  country. 

By  the  "  Expectation  of  Life,"  must  be  understood  the 
average  age  of  any  number  of  individuals.  Thus,  if  100  in- 
fants be  taken,  some  dying  in  infancy,  some  in  childhood, 
some  in  youth,  some  in  middle  life,  and  some  in  old  age,  the 
average  ages  of  all  will  be  38.72  years.  So  from  10  years 
old,  the  average  age  is  48.82  years. 

Quest. — 269*  What  is  an  insurance  ?  270.  What  is  necessary  to  ena- 
ble a  company  to  fix  their  premiums  ?  How  is  the  expectation  determined  ? 
What  Table  is  generally  used  in  this  country?  What  do  you  under- 
stand by  the  expectation  of  life  ? 


LIFE    INSURANCE. 


397 


TABLE    { 

SHOWING    THE 

EXPECTATION    OF    LIFE. 

Age. 

Expectation. 

Age. 

Expectation. 

Age. 

Expectation. 

Age. 

Expectation. 

0 

38.72 

26 

37.14 

52 

19.68 

78 

6.12 

1 

44.68 

27 

36.41 

53 

18.97 

79 

5.80 

2 

47.55 

28 

35.69 

54 

18.28 

80 

5.51 

3 

49.82 

29 

35.00 

55 

17.58 

81 

5.21 

4 

50.76 

30 

34.34 

56 

16.89 

82 

4.93 

5 

51.25 

31 

33.68 

57 

16.21 

83 

4.65 

6 

51.17 

32 

33.03 

58- 

15.55 

84 

4.39 

7 

50.80 

33 

32.36 

59 

14.92 

85 

4.12 

8 

50.24 

34 

31.68 

60 

14.34 

86 

3.90 

9 

45.57 

35 

31.00 

61 

13.82 

87 

3.71 

10 

48.82 

36 

30.32 

62 

13.31 

88 

3.59 

11 

48.04 

37 

29.64 

63 

12.81 

.  89 

3.47 

12 

47.27 

38 

28.96 

64 

12.30 

90 

3.28 

13 

46.51 

39 

28.28 

65 

11.79 

91 

3.26 

14 

45.75 

40 

27.61 

66 

11.27 

92 

3.37 

15 

45.00 

41 

26.97 

67 

10.75 

93 

3.48 

16 

44.27 

42 

26.34 

68 

10.23 

94 

3.53 

17 

43.57 

43 

25.71 

69 

9.70 

95 

3.53 

18 

42.87 

44 

25.09 

70 

9.19 

96 

3.46 

19 

42.17 

45 

24.46 

71 

8.65 

97 

3.28 

20 

41.46 

46 

23.82 

72 

8.16 

98 

3.07 

21 

40.75 

47 

23.17 

73 

7.72 

99 

2.77 

22 

40.04 

48 

22.50 

74 

7.33 

100 

2.28 

23 

39.31 

49 

21.81 

75 

7.01 

101 

1.79 

24 

38.59 

50 

21.11 

76 

6.69 

102 

1.30 

25 

37.86 

51 

20.39 

77 

6.40 

103 

0.83 

271.  From  the  above  table,  and  the  value  of  money,  which 
is  shown  by  the  rate  of  interest,  a  company  can  calculate  with 
great  exactness  the  amount  which  they  should  receive  an- 
nually, for  an  insurance  on  a  life  for  any  number  of  years,  or 
during  its  entire  continuance. 

Among  the  principal  life  insurance  companies  in  the  United 
States,  are  the  New  York  Life  Insurance  and  Trust  Com- 
pany, the  Girard  Life  Insurance,  Annuity,  and  Trust  Com- 
pany of  Philadelphia,  and  the  Massachusetts  Hospital  Life 
Insurance  and  Trust  Company  of  Boston. 

Quest. — Explain  tko  table  showing  the  expectation  of  life.  271.  What 
must  be  known  besides  the  expectation  of  life  in  order  to  find  the  premium? 
What  are  the  principal  life  insurance  companies  in  the  United  States?  How 
do  you  find  t\\e  amount  w}uc!i  must  be  paid  for  the  insurance  of  \ 

13* 


298 


LIFE    INSURANCE. 


NEW  YORK  AND  PHILA.  COMPANIES. 

MASSACHUSETTS. 

Age. 

1  year. 

7  years. 

For  life. 

1  year. 

7  years. 

For  life.   1 

14 

.72 

.86 

1.53 

.89 

1.08 

1.88 

15 

.77 

.88 

1.56 

.90 

1.15 

1.93 

16 

.84 

.90 

1.62 

.96 

1.23 

1.99 

17 

.86 

.91 

1.65 

1.06 

1.30 

2.04 

18 

.89 

.92 

1.69 

1.16 

1.38 

2.09 

19 

.90 

.94 

1.73 

1.25 

1.43 

2.14 

'20 

.91 

.95 

1.77 

1.36 

1.48 

2.18 

21 

.92 

.97 

1.82 

1.44 

1.50 

2.23 

22 

.94 

.99 

1.88 

1.46 

1.53 

2.26 

23 

.97 

1.03 

1.93 

1.49 

1.55 

2.31 

24 

.99 

1.07 

1.98 

1.51 

1.58 

2.35 

25 

1.00 

1.12 

2.04 

1.53 

1.60 

2.40 

26 

1.07 

1.17 

2.11 

1.55 

1.63 

2.45 

27 

1.12 

1.23 

2.17 

1.58 

1.66 

2.50 

28 

1.20 

1.28 

2.24 

1.60 

1.69 

2.55 

29 

1.28 

1.35 

2.31 

1.64 

1.71 

2.61 

30 

1.31 

1.36 

2.36 

1.66 

1.75 

2.66 

31 

1.32 

1.42 

2.43 

1.69 

1.78 

2.73 

32 

1.33 

1.46 

2.50 

1.71 

1.81 

2.79 

33 

1.34 

1.48 

2.57 

1.75 

1.84 

2.85 

34 

1.35 

1.50 

2.64 

1.79 

1.89 

2.93 

35 

1.36 

1.53 

2.75 

1.81 

1.94 

2.99 

36 

1.39 

1.57 

2.81 

1.85 

1.98 

3.06 

37 

1.43 

1.63 

2.90 

1.89 

2.05 

3.14 

38 

1.48 

1.70 

3.05 

1.93 

2.09 

3.23 

39 

1.57 

1.76 

3.11 

1.96 

2.15 

3.31 

40 

1.69 

1.83 

3.20 

2.04 

2.20 

3.40 

41 

1.78 

1.88 

3.31 

2.10 

2.26 

3.49 

42 

1.85 

1.89 

3.40 

2.18 

2.33 

3.59 

43 

1.89 

1.92 

3.51 

2.23 

2.39 

3.69 

44 

1.90 

1.94 

3.63 

2.28 

2.46 

3.79 

45 

1.91 

196 

3.73 

2.34 

2.54 

3.90 

46 

1.92 

a. 98 

3.87 

2.39 

2.63 

4.01 

47 

1.93 

1.99 

4.01 

2.45 

2.71 

4.13 

48 

1.94 

2. 02 

4.17 

2.51 

2.81 

4.25 

49 

1.95 

2-04 

4.49 

2.61 

2.93 

4.39 

50 

1.96 

209 

4.60 

2.75 

3.04 

4.54 

51 

1.97 

2.20 

4.75 

'    2.86 

3.14 

4.68 

52 

2.02 

2.37 

4.90 

2.95 

3.24 

4.83 

53 

2.10 

2.59 

5.24 

3.05 

3.35 

4.98 

54 

2.18 

2.89 

5.49 

3.15 

3.48 

5.14 

55 

2.32 

3.21 

5.78 

3.25 

3.60 

5.31 

56 

2.47 

3.56 

6.05 

3.36 

3.74 

5.50 

57 

2  70 

4.20 

6.27 

3.49 

3.88 

5.70 

58 

3.14 

4.31 

6.50 

3.61 

4.03 

5.91 

59 

3.67 

4.63 

6.75 

3.75 

4.19 

6.14 

bO 

4.35 

4  91 

7.00 

3.90 

4.35 

6.36 

ENDOWMENTS    AND    ANNUITIES.  299 

The  above  table  shows  the  rates  at  which  they  insure 
the  amount  of  $100  for  1  year,  for  7  years,  or  for  life.  It 
should  be  observed,  that  when  a  person  insures  for  7  years 
or  for  life,  he  pays  annually  the  premium  set  opposite  the  age. 
Having  found  the  premium  for  $100,  it  is  easily  found  for 
any  other  amount,  by  simply  multiplying  by  the  amount  and 
dividing  by  100. 

EXAMPLES. 

1.  What  will  be  the  premium  per  annum  on  the  insurance 
of  a  life  for  7  years,  for  $4500,  the  person  being  at  the  age 
of  40  years,  in  the  New  York  or  Philadelphia  companies  ? 

Premium  per  annum  for  7  years  on  $100  =  1,83  ; 
then,  1,83  X  4500  -^  100  =  82,35: 

hence,  $82,35  is  the  premium  per  annum. 

2.  What  would  be  the  premium  per  year  if  insured  for  life  ? 

3.  A  person  at  21  wishes  to  insure  at  his  death  $8500  to 
nis  friends  :  how  much  must  he  pay  per  annum  to  insure  that 
amount  at  his  death,  in  the  Boston  Company  ? 


ENDOWMENTS  AND  ANNUITIES. 

272.  An  Endowment  is  a  certain  sum  to  be  paid  at  the 
expiration  of  a  given  time,  in  case  the  person  on  whose  life 
it  is  taken  shall  live  till  the  expiration  of  the  time  named. 

273.  Annuities  are  certain  annual  or  periodical  payments 
made  to  individuals  by  incorporated  companies  or  associa- 
tions, for  a  given  sum  paid  in  hand. 

274.  The  following  table  shows  the  value  of  an  endow- 
ment purchased  for  $100,  at  the  several  periods  mentioned 
on  the  column  of  ages,  the  endowment  to  be  paid  if  the  per- 
son attains  the  age  of  21  years. 

Quest. — 272.  What  is  an  endowment?  273.  What  is  an  annuity? 
274.  What  does  the  table  of  endowments  show  ? 

13* 


a)0 


ENDOWMENTS    AND    ANNUITIES. 


TABLE    OF    ENDOWMENTS. 


Age. 

Birth  . . . 
3  months 
6       "      . 

9       "      . 

1  year . . . 

2  "     ... 

3  "    ... 

4  "    ... 


Sum  to  be  paid 
at  21,  if  alive. 
$376,84 
344,28 
331,46 
318,90 
306,58 
271,03 
243,69 
225,42 


.  Sum  to  be  paid 

^Se.  at  21,  if  alive, 

5 $210,53 

6 198,83 

7 188,83 

8 179,97 

9 171,91 

10 164,46 

11 157,43 

12 150,64 


A_o  Sum  to  be  paid 

^^®-  at  21,  if  alive. 

13 $144,12 

14 137,86 

15 131,83 

16 125,97 

17 120,31 

18 114,89 

19 109,70 

20 •    104,74 


275.  The  following  table  exhibits  the  sums  which  must  be 
paid,  at  the  several  ages  named,  to  purchase  an  annuity  of 
$100  a  year  in  the  Massachusetts  Life  Insurance  Co.,  and 
in  the  Girard  Life  Insurance,  Annuity,  and  Trust  Company, 
Philadelphia. 


Age. 

20 $1836,30 

21 1823,30 

22 1809,50 

23 1795,10 

24 1780,10 

25 1764,50 

26 1748,60 

27 1732,00 

28 1715,40 

29 1699,70 

30 1685,20 

31 1670,50 

32 1655,20 

33 1639,00 

34 1621,90 

35 1604,10 

36 1585,60 

37 1566,60 

38 1547,10 


Age. 

39 $1527,20 

40 1507,40 

41 1488,30 

42 1469,40 

43 1450,50 

44 1430,80 

45 1410,40 

46 1388,90 

47 1366,20 

48 1341,90 

49.: 1315,30 

50 1300,00 

51.\ 1280,00 

52 1260,00 

53 1240,00 

54 1220,00 

55 1200,00 

56 1175,00 

57 1150,00 


Age. 

58 $1125,00 

59 1100,00 

60 1070,00 

61 1045,00 

62 1020,00 

63 995,00 

970,00 


10. 

11  66. 

II  67. 
il68. 
!!69. 

il70. 

;|7i. 

i|72. 

ii73. 
i|74. 
'|75. 


940,00 
910,00 
880,00 
850,00 

i5o,oo 

790,00 
780,00 
770,00 
760,00 
750,00 
740,00 


EXAMPLES. 

1.  What  sum  at  birth  will  purchase  an  endowment  at  21 
of  $859,61  . 

2.  What  sum  at  the  age  of  30  years  will  purchase  an  an 
nuity  of  $3150? 

Quest. — ^275.  What  does  the  table  of  annnities  show  ? 


COINS    AND    CURRENCIES.  30l 


COINS  AND  CURRENCIES. 

276.  Coins  are  pieces  of  metal,  of  gold,  silver,  or  copper, 
of  fixed  values,  and  impressed  with  a  public  stamp  prescribed 
b}^  the  country  where  they  are  made.  These  are  called 
specie,  and  are  generally  declared  to  be  a  legal  tender  in 
payment  of  debts.  The  Constitution  of  the  United  States 
provides,  that  gold  and  silver  only  shall  be  a  legal  tender. 

The  coins  of  a  country  and  those  of  foreign  countries 
having  a  fixed  value  established  by  law,  together  with  bank 
notes  redeemable  in  specie,  make  up  what  is  called  the 
Currency, 

277-  A  foreign  coin  may  be  said  to  have  four  values  : 

1st.  The  intrinsic  value,  which  is  determined  by  the 
amount  of  pure  metal  which  it  contains. 

2d.  The  custom  house  or  legal  value,  which  is  fixed  Jby 
law. 

3d.  The  mercantile  value,  which  is  the  amount  it  will  sell 
for  in  open  market. 

4th.  The  exchange  value,  which  is  the  value  assigned  to 
it  in  buying  and  selling  bills  of  exchange  between  one  coun- 
try and  another. 

Let  us  take,  as  an  example,  the  English  pound  sterling, 
which  is  represented  by  the  gold  sovereign.  Its  intrinsic 
value,  as  determined  at  the  Mint  in  Philadelphia,  compared 
with  our  gold  eagle,  is  $4,861.  Its  legal  or  custom  house 
value  is  $4,84.  Its  commercial  value,  that  is,  what  it  will 
bring  in  Wall  street,  New  York,  varies  from  $4,83  to  $4,86, 
seldom  reaching  either  the   lowest  or  highest  limit.     The 

Quest. — 276.  What  are  coins  ?  What  are  they  called  ?  What  is  de- 
clared in  regard  to  them  ?  What  is  provided  by  the  Constitution  of  the 
United  States?  What  do  you  understand  by  Currency?  277.  How  many 
values  may  a  coin  be  said  to  have  ?  What  is  the  intrinsic  value  ?  What 
is  the  mercantile  value  ?     What  is  the  exchange  value  ? 


302 


COINS    AND    CURRENCIES. 


exchange  value  of  the  English  pound,  is  $4,44 1,  and  was  the 
legal  value  before  the  change  in  our  standard.  This  change 
raised  the  legal  value  of  the  pound  to  $4,84,  but  merchants 
and  dealers  in  exchange  preferred  to  retain  the  old  value, 
which  became  nominal,  and  to  add  the  difference  in  the  form 
of  a  premium  on  exchange,  which  is  explained  in  Art.  292. 

TABLE    OF    FOREIGN    COINS    WHOSE    VALUES    ARE    FIXED 
BY    LAW. 


Franc  of  France  and  Belgium 

Florin  of  the  Netherlands 

Guilder  of  do.  

Livre  Tournois  of  France 

Milrea  of  Portugal 

Milrea  of  Madeira 

Milrea  of  the  Azores 

Marc  Banco  of  Hamburg 

Pound  SterHng  of  Great  Britain 

Pagoda  of  India 

Real  Vellon  of  Spain 

R^al  Plate  of      do 

Rupee  Company 

Rupee  of  British  India 

Rix  Dollar  of  Denmark 

Rix  Dollar  of  Prussia 

Rix  Dollar  of  Bremen 

Rouble,  silver,  of  Russia 

Tale  of  China 

Dollar  of  Sweden  and  Norway 

Specie  Dollar  of  Denmark 7*. 

Dollar  of  Prussia  and  Northern  States  of  Germany .... 

Florin  of  Southern  States  of  Germany 

Florin  of  Austria  and  city  of  Augsburg 

Lira  of  the  Lombardo  Venetian  Kingdom 

Lira  of  Tuscany 

Lira  of  Sardinia 

Ducat  of  Naples 

Ounce  of  Sicily 

Pound  of  Nova  Scotia,  New  Brunswick,  Newfoundland, 
and  Canada 


15 


cts. 
18 
40 
40 
181 
12 
00 

83^3 

35 

84 

84 

05 

10 

441 

44-5 

00 

68J 

78| 

75 

48 

06 

05 


40 
481 
16 
16 

80 
40 

04 


Quest. — Give  the  different  values  of  the  English  sovereign.  How  came 
the  value  of  the  sovereign  to  be  altered?  How  is  the  difference  now 
made  up  ? 


EXCHANGE. 


303 


TABLE    OF    FOREIGN    COINS    WHOSE    VALUES    ARE    FIXED 
BY    USAGE, 

When  a  Consular's  certificate  of  the  real  value  or  rate  of  ex- 
change is  not  attached  to  the  invoice. 


Berlin  Rix  Dollar 

Current  Marc 

Crown  of  Tuscany 

Elberfpldt  Rix  Dollar 

Florin  of  Saxony 

"  Bohemia 

Elberfeldt 

"  Prussia 

Trieste 

"  Nuremburg 

"  Frankfort 

Basil 

St.  Gaul 

"  Creveld. 

Florence  Livre 

Genoa         do 

Geneva       do 

Jamaica  Pound -. 

Leghorn  Dollar 

Leghorn  Livre  (6 J  to  the  dollar) . 

Livre  of  Catalonia 

Neufchatel  Livre 

Pezza  of  Leghorn 

Rhenish  Rix  Dollar 

Swiss  Livre 

Scuda  of  Malta 

Turkish  Piastre 


cts, 

69i 

28" 

05 

69| 

48 

48 

40 

22j 

48 

40 

40 

41 

40t^A 

40 

15 

18f 

21 

00 

90 

261 

90 

60j 

27 

40 

05 


[The  above  Tables  are  taken  from  a  work  on  the  Tariff,  by  E.  D 
Ogden,  Esq.,  of  the  New  York  Custom  House.] 


EXCHANGE. 

278.  Exchange  is  a  term  which  denotes  the  payment  of 
money  by  a  person  residing  in  one  place  to  a  person  residing 
in  another.  The  payment  is  generally  made  by  means  of  a 
bill  of  exchange. 

Quest. — 278  What  is  exchange?  How  is  the  payment  generally 
made? 


304  EXCHANGE. 

279.  A  Bill  of  Exchange  is  an  open  letter  of  request 
from  one  person  to  another,  desiring  the  payment  to  a  third 
party  named  therein,  of  a  certain  sum  of  money  to  be  paid  at 
a  specified  time  and  place.  There  are  always  three  parties 
to  a  bill  of  exchange,  and  generally  four. 

1.  He  who  writes  the  open  letter  of  request,  is  called  the 
drawer  or  maker  of  the  bill. 

2.  The  person  to  whom  it  is  directed  is  called  the  drawee, 

3.  The  person  to  whom  the  money  is  ordered  to  be  paid 
is  called  the  payee ;  and 

4.  Any  person  who  purchases  a  bill  of  exchange  is  called 
the  buyer  or  remitter. 

280.  Bills  of  exchange  are  the  proper  money  of  commerce. 
Suppose  Mr.  Isaac  Wilson  of  the  city  of  New  York,  ships 
1000  bags  of  cotton,  worth  £96000,  to  Samuel  Johns  &  Co. 
of  Liverpool ;  and  at  about  the  same  time  William  James  of 
New  York^  orders  goods  from  Liverpool,  of  Ambrose  Spooner, 
to  the  amount  of  eighty  thousand  pounds  sterling.  Now,  Mr. 
Wilson  draws  a  bill  of  exchange  on  Messrs.  Johns  &l  Co.  in 
the  following  form  :  viz., 


Exchange  for  £80000.  New  York,  July  30th,  1846. 

Sixty  days  after  sight  of  this  my  first  Bill  of  Exchange 
(second  and  third  of  the  same  date  and  tenor  unpaid*)  pay 
to  David  C.  Jones  or  order,  eighty  thousand  pounds  sterling, 
with  or  without  further  advice.  Isaac  Wilson. 

Messrs.  Samuel  Johns  &  Co.,  \ 
Merchants,  Liverpool. 

Let  us  now  suppose  that  Mr.  James  purchases  this  bill  of 
David  C.  Jones  for  the  purpose  of  sending  it  to  Ambrose 

*  Three  bills  are  generally  drawn  for  the  same  amount,  called  the  fii-st, 
second,  and  third,  and  together  they  form  a  set.  One  only  is  paid,  and 
then  the  other  two  are  of  no  value.  This  arrangement  avoids  the  acci- 
dents and  delays  incident  to  transmitting  the  bills. 

Quest. — 279.  What  is  a  bill  of  exchange  ?  How  many  parties  are  there 
to  a  bill  of  exchange  ?  Name  them.  280.  How  do  bills  of  exchange  aid 
commerce  ?     Name  all  the  parties  of  the  bill  in  this  example. 


EXCHANGE  305 

Spooner  of  Liverpool,  whom  he  owes.  We  shall  then  have 
all  the  parties  tp  a  bill  of  exchange ;  viz.,  Isaac  Wilson,  the 
maker  or  drawer ;  Messrs.  Johns  &  Co.,  the  drawees ;  David 
C.  Jones,  the  payee;  and  William  James,  the  buyer  or  re- 
mitter. 

281.  A  bill  of  exchange  is  called  an  inland  bill,  when  the 
drawer  and  drawee  both  reside  in  the  same  country;  and 
when  they  reside  in  different  countries,  it  is  called  a  fareign 
bill.  Thus,  all  bills  in  which  the  drawer  and  drawee  reside 
in  the  United  States,  are  inland  bills  ;  but  if  one  of  them  re- 
sides in  England  or  France,  the  bill  is  a  foreign  bill. 

282.  The  time  at  which  a  bill  is  made  payable  varies,  and 
is  a  matter  of  agreement  between  the  drawer  and  buyer. 
They  may  either  be  drawn  at  sight,  or  at  a  certain  number 
of  days  after  sight,  or  at  a  certain  number  of  days  after  date, 

283.  Days  of  Grace  are  a  certain  number  of  days  grant- 
ed to  the  person  who  pays  the  bill,  after  the  time  named  in 
the  bill  has  expired.  In  the  United  States  and  Great  Britain 
three  days  are  allowed. 

284.  In  ascertaining  the  time  when  a  bill  payable  so  many 
days  after  sight,  or  after  date,  actually  falls  due,  the  day  of 
presentment,  or  the  day  of  the  date,  is  not  reckoned.  When 
the  time  is  expressed  in  months,  calendar  months  are  always 
understood. 

If  the  month  in  which  a  bill  falls  due  is  shorter  than  the 
one  in  which  it  is  dated,  it  is  a  rule  not  to  go  on  into  the  next 
month.  Thus  a  bill  drawn  on  the  28th,  29th,  30th,  or  31st 
of  December,  payable  two  months  after  date,  would  fall  due 

Quest.— 281.  What  is  an  inland  bill  ?  What  is  a  foreign  bill?  Are  bills 
drawn  between  one  state  and  another  inland  or  foreign  ?  282.  How  is  the 
time  determined  at  which  a  bill  is  made  payable  ?  How  are  bills  always 
drawn  ?  283.  What  are  days  of  grace  ?  How  many  days  of  grace  are 
allowed  in  this  country  and  in  Great  Britain  ?  284.  In  ascertaining  the 
time  when  a  bill  is  payable,  what  days  are  reckoned  ?  When  the  time  is 
expressed  in  months,  what  kind  of  months  is  understood  ?  If  the  month  in 
which  the  bill  falls  due  is  shorter  than  that  in  which  it  is  drawn,  what  rule 
is  observed? 


306  EXCHANGE. 

on  the  last  of  February,  except  for  the  days  of  grace,  and 
would  be  actually  due  on  the  third  of  March. 

ENDORSING    BILLS. 

285.  In  examining  the  bill  of  exchange  drawn  by  Isaac  Wil- 
son, it  will  be  seen  that  Messrs.  Johns  &  Co.  are  requested 
to  pay  the  amount  to  David  C.  Jones  or  order ;  that  is,  either 
Mr.  Jones  or  to  any  other  person  named  by  him.  If  Mr. 
Jones  simply  writes  his  name  on  the  back  of  the  bill,  he  is 
said  to  endorse  it  in  blank,  and  the  'drawees  must  pay  it  to 
any  rightful  owner  who  presents  it.  Such  rightful  owner  is 
called  the  holder,  and  Mr.  Jones  is  called  the  endorser. 

If  Mr.  Jones  writes  on  the  back  of  the  bill,  over  his  signa- 
ture, "  Pay  to  the  order  of  William  James,"  this  is  called  a 
special  endorsement,  and  William  James  is  the  endorsee,  and 
he  may  either  endorse  in  blank  or  write  over  his  signature 
**  Pay  to  the  order  of  Ambrose  Spooner,"  and  the  drawees, 
Messrs.  Johns  &  Co.,  will  then  be  bound  to  pay  the  amount 
to  Mr.  Spooner. 

A  bill  drawn  payable  to  bearer,  maybe  transferred  by  mere 
delivery. 

ACCEPTANCE. 

286.  When  the  bill  drawn  on  Messrs.  Johns  &  Co.  is 
presented  to  them,  they  must  inform  the  holder  whether  or 
not  they  will  pay  it  at  the  expiration  of  the  time  named. 
Their  agreement  to  pay  it  is  signified  by  writing  across  the 
face  of  the  bill,  and  over  their  signature  the  word  "  accept- 
ed," and  they  are  then  called  the  acceptors, 

LIABILITIES    OF    THE    PARTIES. 

287.  The  drawee  of  a  bill  does  not  become  responsible  lor 
its  payment  until  after  he  has  accepted.     On  the  presenta- 

Quest. — 285.  What  is  an  endoi-sement  in  blank  ?  What  is  the  person 
making  it  called  ?  What  is  a  special  endorsement  ?  What  is  the  effect 
of  an  endorsement?  How  may  a  bill  drawn  to  bearer  be  transferred? 
286.  What  is  an  acceptance  ?  How  is  it  made  ?  287.  When  does  the 
^  iwee  of  a  bill  become  respoiosiblo  for  its  payment  ? 


EXCHANGE.  307 

tion  of  the  bill,  if  the  drawee  does  not  accept,  the  holder 
should  immediately  take  means  to  have  the  drawer  and  all 
the  endorsers  notified.  Such  notice  is  called  a  protest,  and 
is  given  by  a  public  officer  called  a  notary,  or  notary  public. 
If  the  parties  are  not  notified  in  a  reasonable  time,  they  are 
not  responsible  for  the  payment  of  the  bill. 

If  the  drawer  accepts  the  bill  and  fails  to  make  the  pay- 
ment when  it  becomes  due,  the  parties  must  be  notified  as 
before,  and  this  is  called  protesting  the  hill  for  non-payment. 
If  the  endorsers  are  not  notified  in  a  reasonable  time,  thoy 
are  not  responsible  for  the  amount  of  the  bill. 

PAR  OF  EXCHANGE COURSE  OF  EXCHANGE. 

288.  The  intrinsic  par  of  exchange,  is  a  term  used  to  com- 
pare the  coins  of  different  countries  with  each  other,  with 
respect  to  their  intrinsic  values,  that  is,  with  reference  to  the 
amount  of  pure  metal  in  each.  Thus,  the  English  sovereign, 
which  represents  the  pound  sterling,  is  intrinsically  worth 
$4,861  in  our  gold,  taken  as  a  standard,  as  determined  at  the 
Mint  in  Philadelphia.  This,  therefore,  is  the  value  at  which 
the  sovereign  must  be  reckoned,  in  estimating  the  par  of  ex- 
change. 

289.  The  commercial  par  of  exchange  is  a  comparison  of 
the  coins  of  different  countries  according  to  their  market 
value.  Thus,  the  market  value  of  the  English  sovereign, 
varying  from  $4,83  to  $4,85  (Art.  277),  the  commercial  par 
of  exchange  will  fluctuate.  It  is,  however,  always  deter- 
mined when  we  know  the  value  at  which  the  foreign  coin 
sells  in  our  market. 

Quest. — If  the  drawee  does  not  accept,  what  must  the  holder  do  ?  What 
is  such  notice  called  ?  By  whom  is  it  made  ?  If  the  parties  to  the  bill 
are  not  notified,  what  is  the  consequence  ?  Jf  the  drawee  accepts  the  bill 
and  fails  to  make  the  payment,  what  must  then  be  done  ?  If  the  bill  is 
not  protested,  what  will  be  the  consequence  ?  288.  What  do  you  understand 
by  the  intrinsic  par  of  exchange  ?  What  is  the  intrinsic  value  of  the  Eng- 
lish sovereign?  289.  What  is  the  commercial  par  of  exchange?  What 
is  the  commercial  value  of  the  English  sovereign  ? 


308  EXCHANGE. 

290.  The  course  of  exchange  is  tlie  variable  price  which  is 
paid  at  one  place  for  bills  of  exchange  drawn  on  another. 
The  course  of  exchange  differs  from  the  intrinsic  par  of  ex- 
change, and  also  from  the  commercial  par,  in  the  same  way 
that  the  market  price  of  an  article  differs  from  its  natural 
price.  The  commercial  par  of  exchange  would  at  all- times 
determine  the  course  of  exchange,  if  there  were  no  fluctua- 
tions in  trade. 

291.  When  the  market  price  of  a  foreign  bill  is  above  the 
commercial  par,  the  exchange  is  said  to  be  at  a  premium,  or 
in  favor  of  the  foreign  place,  because  it  indicates  that  the 
foreign  place  has  sold  more  than  it  has  bought,  and  that 
specie  must  be  shipped  to  make  up  the  difference.  When 
the  market  price  is  below  this  par,  exchange  is  said  to  be 
below  par,  or  in  fav^  of  the  place  where  the  bill  is  drawn. 
Such  place  will  then  be  a  creditor,  and  the  debt  must  be  paid 
in  specie  or  other  property.  It  should  be  observed  that  a 
favorable  state  of  exchange  is  advantageous  to  the  buyer  but 
not  to  the  seller,  whose  interest,  as  dealer  in  exchange,  is 
identified  with  that  of  the  place  on  which  the  bill  is  drawn. 

292.  It  was  stated  in  Art.  277  that  the  exchange  value  of 
the  pound  sterling  is  f  4,44|-  =  4,4444+  ;  that  is,  this  value 
is  the  basis  on  which  the  bills  of  exchange  are  drawn.  Now 
this  value  being  below  both  the  commercial  and  intrinsic 
value,  the  drawers  of  bills  increase  the  course  of  exchange 
so  as  to  make  up  this  deficiency. 

For  example,  if  we  add  to  the  exchange  value  of  the  pound, 
9  per  cent,  we  shall  have  its  commercial  value,  very  nearly. 
Thus,  exchange  value       -       -       -     =  $4,4444-}- 

Nine  per  cent = ,3999  + 

which  gives $4,l5443 

Quest. — 290.  What  do  you  understand  by  the  course  of  exchange? 
How  does  it  differ  from  the  intrinsic  par  and  the  commercial  par  ?  What 
causes  it  to  differ  from  the  commercial  par?  291.  What  is  said  when  the 
price  of  a  foreign  bill  is  above  the  commercial  par?  When  it  is  bnlow  it? 
To  whom  is  a  favorable  state  of  exchange  advantageous?  To  whom  is  it 
injurious  ?  .  292.  What  is  the  exchange  value  of  the  pound  sterling  ? 


EXCHANGE.  309 

and  this  is  the  average  of  the  commercial  value,  very  nearly. 
Therefore,  when  the  course  of  exchange  is  at  a  premium  of 
9  per  cent,  it  is  at  the  commercial  par,  and  as  between  Eng- 
land and  this  country  it  would  stand  near  this  point,  but  for 
the  fluctuations  of  trade  and  other  accidental  circumstances. 

INLAND    BILLS. 

293.  We  have  seen  that  inland  bills  are  those  in  which 
the  drawer  and  drawee  both  reside  in  the  same  country 
(Art.  281). 

EXAMPLES. 

1.  A  merchant  at  New  Orleans  wishes  to  remit  to  New 
York  $8465,  and  exchange  is  1 J  per  cent  premium.  How 
much  must  he  pay  for  such  a  bill  ? 

2.  A  merchant  in  Boston  wishes  to  pay  in  Philadelphia 
$8746,50 ;  exchange  between  Boston  and  Philadelphia  is 
li  per  cent  below  par.     What  must  he  pay  for  a  bill  ? 

3.  A  merchant  in  Philadelphia  wishes  to  pay  $9876,40  in 
Baltimore,  and  finds  exchange  to  be  1  per  cent  below  par  • 
what  must  he  pay  for  the  bill  ? 

ENGLAND. 

294.  It  has  already  been  stated  that  the  exchanges  be- 
tween this  country  and  England  are  made  in  pounds,  shillings 
and  pence,  and  that  the  exchange  value  of  the  pound  sterling 
is  $4,44|^,  and  that  the  premiums  are  all  reckoned  from  this 
standard. 

EXAMPLES. 

1.  A  merchant  in  New  York  wishes  to  remit  to  Liverpool 
£1167  10^.  6 J.,  exchange  being  at  8^  per  cent  premium. 
How  much  must  he  pay  for  the  bill  in  Federal  money  ? 

Quest.— 293.  What  are  inland  bills?  294.  In  what  currency  are  the 
exchanges  between  this  country  and  England  made "?  What  is  the  ex- 
change value  of  the  pound  sterling  ? 


310  EXCHANGE. 

First,  £1167  10^.  6cZ.        -  -  =  £1167.523 

For  81  per  cent  multiply  by  -  .085 

the  product  is  the  premium  -  =         99.2396S5 

this  being  added  gives        -  -  £1266.764625 

which  reduced  to  dollars  and  cents  at  the  rate  of  $4,44|  to  the 
pound,  gives  the  amount  which  must  be  paid  for  the  bill  in 
dollars  and  cents. 

2.  A  merchant  has  to  remit  £36794  8^.  9d.  to  London, 
how  much  must  he  pay  for  a  bill  in  dollars  and  cents,  ex 
change  being  7|^  per  cent  premium  ? 

3.  A  merchant  in  New  York  wishes  to  remit  to  London 
$67894,25,  exchange  being  at  a  premium  of  9  per  cent. 
What  will  be  the  amount  of  his  bill  in  pounds,  shillings  and 
pence  ? 

Note. — ^Add  the  amount  of  the  premium  to  the  exchange  value 
of  the  pound,  viz.  $4,44^^,  which  in  this  case  gives  $4,84443,  and 
then  divide  the  amount  in  dollars  by  this  sum,  and  the  quotient  will 
be  the  amount  of  the  bill  in  pounds  and  the  decimal  of  a  pound. 

4.  A  merchant  in  New  York  owes  £1256  18^.  9d.  in  Lon- 
don ;  exchange  at  a  nominal  premium  of  7J  per  cent :  how 
much  money  in  Federal  currency  will  be  necessary  to  pur- 
chase the  bill  ? 

5.  I  have  $947,86  and  wish  to  remit  to  London  £364 
18^.  8c?.,  exchange  being  at  8^  per  cent:  how  much  addi- 
tional money  will  be  necessary  ? 

FRANCE. 

295.  The  accounts  in  France,  and  the  exchange  between 
France  and  other  countries,  are  all  kept  in  francs  and  cen- 
times, which  are  hundredths  of  the  franc.  We  see  from  the 
table  that  the  value  of  the  franc  is  18.6  centi^  which  gives 
very  nearly,  5  francs  and  38  centimes  to  the  dollar.  The 
rate  of  exchange  is  computed  on  the  value  18.6  cents,  but  is 
often  quoted  by  stating  the   value  of  the  dollar  in  francs. 

Quest. — 295.  In  what  currency  are  the  exchanges  with  France  con- 
ducted?    What  is  a  centime?     What  is  the  value  of  a  franc? 


EXCHANGE.  311 

Thus,  exchange  on  Paris  is  said  to  be  5  francs,  40  centimes, 
that  is,  one  dollar  will  buy  a  bill  on  Paris  of  5  francs  and  40 
hundredths  of  a  franc. 

EXAMPLES. 

1.  A  merchant  in  New  York  wishes  to  remit  167556  francs 
to  Paris,  exchange  being  at  a  premium  of  IJ  per  cent. 
What  will  be  the  cost  of  his  bill  in  dollars  and  cents  ? 

Commercial  value  of  the  franc     -       -       18.6  cents 

Add  1^  per  cent 279 

Gives  value  for  remitting       -       -       -       18.879  cents; 
then,     167556  X  18.879  =  $31632,89724, 
which  is  the  amount  to  be  paid  for  the  bill. 

2.  What  amount  in  dollars  and  cents  will  purchase  a  bill 
on  Paris  for  86978  francs,  exchange  being  at  the  rate  of  5 
francs  and  2  centimes  to  the  dollar  ? 

First,      86978  -^  5.02  =  $17326,274  +  the  amount. 
Is  this  bill  above  or  below  par  ?     What  per  cent  1 

3.  How  much  money  must  be  paid  to  purchase  a  bill  of 
exchange  on  Paris  for  68097  francs,  exchange  being  3  per 
cent  below  par  ? 

4.  A  merchant  in  New  York  wishes  to  remit  $16785,25 
to  Paris ;  exchange  gives  5  francs  4  centimes  to  the  dollar : 
how  much  can  he  remit  in  the  currency  of  Paris  ? 

HAMBURG. 

296.  Accounts  and  exchanges  with  Hamburg  are  generally 
made  in  the  marc  banco,  valued,  as  we  see  in  the  table,  at 
35  cents. 

EXAMPLES. 

1 .  What  amount  in  dollars  and  cents  will  purchase  a  bill 
of  exchange  on  Hamburg  for  18649  marcs  banco,  exchange 
being  at  2  per  cent  premium  ] 

Quest. — What  is  meant  when  exchange  on  Paris  is  quoted  at  5  francs 
40  centimes?  296.  In  what  are  accounts  kept  at  Hamburg?  What  is 
the  value  of  the  marc  banco  ? 


312  ARBITRATION    OF   EXCHANGE. 

2.  What  amount  will  purchase  a  bill  for  3678  marcs  banco 
reckoning  the  exchange  value  of  the  marc  banco  at  34  cents  ? 
Will  this  be  above  or  below  the  par  of  exchange  ? 


ARBITRATION  OF  EXCHANGE. 

297.  Arbitration  of  exchange  is  the  method  by  which  the 
currency  of  one  country  is  changed  into  that  of  another, 
through  the  medium  of  one  or  more  intervening  currencies, 
with  which  the  first  and  last  are  compared. 

98.  When  there  is  but  one  intervening  currency  it  is 
called  simple  arbitration ;  and  when  there  is  more  than  one 
it  is  called  compound  arbitration.  The  method  of  performing 
this  is  called  the  Chain  Rule. 

299.  The  principle  involved  in  arbitration  of  exchange  is 
simply  this :  To  pass  from  one  system  of  values  through 
several  others,  and  find  the  true  proportion  or  relation  between 
the  first  and  last.  For  example,  suppose  we  wish  to  exchange 
109150  pence  into  dollars  by  first  changing  them  into  shil- 
lings, then  into  pounds,  and  then  into  dollars.  For  this  we 
have, 

12  :  109150  :  :  Is.  :  109150  X  j^^  =  number  of  shillings. 
20  :  109150x^3^  :  :  £l  :  109150XiVXyo=^<^- of  pounds. 
£1:  $4,444:  :  109150XiVx^  •  109150  x^gX^X  ^-^^i 
hence  the  Chain  Rule  may  be  stated  as  follows : 

Multiply  the  sum  to  be  remitted  by  the  following  quotients^ 
after  having  cancelled  the  common  factors,  viz.,  by  a  certain 
amount  at  the  second  place  divided  by  its  equivalent  at  the  first ; 
a  certain  amount  at  the  third  place  by  its  equivalent  at  the  sec- 
ond ;  a  certain  amount  at  the  fourth  place  divided  by  its  equiva- 
lent at  the  third,  and  so  on  to  the  last  place. 

Quest. — 297.  What  is  arbitration  of  exchange?  298.  When  there  is 
but  one  intervening  currency,  what  is  the  exchange  called  ?  When  there 
is  more  than  one,  what  is  it  called  ?  299.  What  principle  is  involved  in 
the  arbitration  of  exchange  1    What  is  the  Chain  Rule  ?     Give  the  rule 


ARBITRATION    OF    EXCHANGE.  313 

Note. — In  the  above  rule  the  amounts  named  are  supposed  to  be 
expressed  in  the  currency  of  the  place  from  which  the  remittance 
is  made.  If  in  any  case  an  amount  is  expressed  in  the  currency  of 
the  place  to  which  the  remittance  is  made,  the  terms  of  the  corre- 
sponding multiplier  must  be  inverted.  The  example  wrought  above 
may  be  thus  stated:  Required  to  transmit  109150  pence  to  a  sec- 
ond place  where  one  piece  of  coin  is  worth  12  at  the  first  place ; 
thence  to  transmit  it  to  a  third  where  one  piece  is  worth  20  at  the 
second ;  thence  to  a  fourth  place  where  4.444  pieces  are  equal  to  1 
at  the  third. 

EXAMPLES. 

1 .  A  merchant  wishes  to  remit  $4888,40  from  New  York  to 
London,  and  the  exchange  is  10  per  cent.  He  finds  that  he 
can  remit  to  Paris  at  5  francs  15  centimes  to  the  dollar,  and 
to  Hamburg  at  35  cents  per  marc  banco.  Now,  the  exchange 
between  Paris  and  London  is  25  francs  80  centimes  for  £1 
sterling,  and  between  Hamburg  and  London  13|  marcs  banco 
for  £i  sterling.     How  had  he  better  remit? 

1st.    To  London  direct. 

The  amount  to  be  remitted  is  $4888,40.  The  exchange 
value  of  £1  is  $4,444,  and  since  the  exchange  is  at  a  premium 
of  10  per  cent,  the  value  of  £1  is  $4,444 +  ,4444 =$4,8884  : 
hence, 

$4888,40  X  4,^-g-4  =  £1000: 
hence,  if  he  remits  direct  he  will  obtain  a  bill  for  £1000. 

2d.  Exchange  through  Paris. 
1.03 
4888,40  X^j^X  — ^^  =  £975,7852  =  £975  15^.  S^d, 

5J6 

Since  5,15  francs  are  equal  to  1  dollar,  the  first  multiplier 

will  be  this  amount  divided  by  $1  ;  and  since  £1  is  equal  to 

25  80  francs,  the  second  multiplier  will  be  £1  divided  by  this 

amount.     Then  by  dividing  by  5  and   multiplying,  we  find 

that  the  amount  remitted  by  the  second  method  would  be 

£975  155.  S^d. 

14 


314  ARBITRATION    OF    EXCHANGE. 

3(1 .  Method  through  Hamburg. 

$4888,40  X  .gV  X  Y3V5  =  1015.771  =  £1015  15.y.  5rf. 
Since  1  marc  banco  is  equal  to  35  cents,  it  is  35  hun 
dredths  of  a  dollar :  hence,  the  first  muhiplier  is  1  marc  banco 
divided  by  .35,  and  the  second  1  divided  by  13.75.  The  re- 
sult shows  that  the  best  w^ay  to  remit  is  through  Hamburg, 
the  next  best  direct,  and  the  most  unfavorable  through  Paris. 
2.  A  merchant  in  London  has  sold  goods  in  Amsterdam  to 
the  amount  of  824  pounds  Flemish,  which  could  be  remitted 
to  London  at  the  rate  of  34^.  Ad.  Flemish  per  pound  sterling. 
He  orders  it  to  be  remitted  circuitously  at  the  following  rates, 
viz.,  to  France  at  the  rate  of  ASd.  Flemish  per  crown ;  thence 
to  Vienna  at  100  crowns  for  60  ducats  ;  thence  to  Hamburg  at 
100 J.  Flemish  per  ducat ;  thence  to  Lisbon  at  50 J.  Flemish 
per  crusado  of  400  reas  ;  and  lastly,  from  Lisbon  to  England 
at  5^.  Sd.  per  milrea :  does  he  gain  or  lose  by  the  circular 
exchange  1 

48J.  Flemish  =  1  crown, 

100  crowns  =  60  ducats, 

1  ducat  =  lOOd,  Flemish, 

50J.  Flemish  =  400  reas, 

1  milrea  or  1000  reas  =  QSd,  sterling. 
/0  ^         17 

824  X  i  X  "^  X^  X^X.  ^^  -  824X  17  _  14008 


^'^^"S^S^"    1     ^^'"^^~       25       -    25 

25 
=  £560  65.  4|J. 
The  direct  exchange  would  give, 

«""  X  34..  4/Flemish  =  824  X  H^  =:  £480  sterHng. 
Hence,  the  amount  gained  by  circuitous  exchange  would 
be  £80  6^.  4|J. 


DUODECIMALS.  315 


DUODECIMALS. 

300.  Duodecimals  are  denominate  fractions  in  which  1 
foot  is  the  unit  that  is  divided. 

The  unit  1  foot  is  first  supposed  to  be  divided  into  12 
equal  parts,  called  inches  or  primes,  and  marked  \ 

Each  of  these  parts  is  supposed  to  be  again  divided  into 
12  equal  parts,  called  seconds,  and  marked  ^\ 

Each  second  is  divided,  in  like  manner,  into  12  equal  parts, 
called  thirds,  and  marked  ^^\ 
This  division  of  the  foot  gives 

V    inch  or  prime     -     -     -     r:=    y^^    of  a  foot. 

V^  second  is  ==  ^^  of  ^^  -     =  y^^  of  a  foot. 

V^^  third  is  =  j3^  of  j^-.  of  j2^  r=  TyVs  ^^  ^  ^'^^*- 

Hence,  in  duodecimals,  the  divisions  Of  the  foot  increase 

from  the  lower  denominations  to  the  higher,  according  to  the 

scale  of  twelves. 

301.  Duodecimals  are  added  and  subtracted  like  other  de- 
nominate numbers,  12  of  a  lesser  denomination  making  one 
of  a  greater,  as  in  the  following 

TABLE. 

12^^^     make    V^     second. 

V^''         "         V      inch  or  prime. 

12^  "         1       foot. 

EXAMPLES. 

1.  In  185^,  how  many  feet?  Ans,  

2.  In  250^^,  how  many  feet  and  inches  ?  Ans.  

3    In  4367''^^  how  many  feet?  Ans.  

4.  In  847^^,  how  many  feet  ?  Ans,  


Quest. — 300.  In  Duodecimals,  what  is  the  unit  that  is  divided  ?  How 
is  it  divided  ?  How  are  these  parts  again  divided  ?  What  ai'e  the  parts 
called?  301.  How  are  duodecimals  added  and  subtracted?  How  many 
of  one  denomination  make  1  of  the  next  greater  ? 


316 


MULTIPLICATION    OF    DUODECIMALS. 


EXAMPLES    IN    ADDITION    AND    SUBTRACTION. 

1.  What  is  the  sum  of  3/Jj.  6'  3''  2'''  and  2ft.  V  W  IV  ? 

2.  What  is  the  sum  of  8ft.  9'  T'  and  6/iJ.  T  3''  4'''  ? 

3.  What  is  the  difference  between  9ft.  3'  5''  6'''  and  7ft. 
3/  (J//  7///  ? 

4.  What  is  the  difference  between  40ft.  6'  6''  and  I9ft.  T'' ? 

5.  What  is  the  sum  of  18/^.  9'  W  5'''  and  I7ft.  6'  T'  ? 

6.  What  is  the  difference  between  21ft.  T''  and  Aft.  9" 
IQU  9///? 

MULTIPLICATION  OF  DUODECIMALS. 

302.  It  is  known  that  feet  multiplied  by  feet  give  square 
feet  in  the  product.  It  is  now  required  to  show  what  frac- 
tions of  the  square  foot  will  arise  from  multiplying  feet  by 
the  divisions,  of  the  foot,  and  the  divisions  of  the  foot  by 
each  other. 

EXAMPLES. 

1.  Multiply  6ft.  r  S''  by  2ft.  9\ 


OPERATION. 

6  T 
2  9' 


W 


Set  down  the  multiplier  under 
the  multiplicand,  so  that  feet  shall 
fall  under  feet,  and  the  correspond- 
ing divisions  under  each  other.  It 
is  found  most  convenient  to  begin 
with  the  highest  denomination  of 
the  multiplier,  and  multiply  it  by 
the  lowest  denomination  of  the  mul- 
tiplicand. Recollecting  that  7^  ex- 
presses ^-^  of  a  foot,  and  that  8^'' 
expresses  ^  of  -^^  of  a  foot,  we 
see  that  2  x  8^^  will  give  1 6-twelfths  of  twelfths  of  a  square 
foot;  that  is,  one-twelfth  and  four  twelfths  of  one  twelfth, 
or  4^\  The  2  feet  multiplied  by  7^  give  14  'weUths  of  a 
square  foot ;  that  is,  1  square  foot  and  two  vwelfths,  or  2\ 
The  feet  multiplied  by  6  give  12  square  fee?. 

Quest. — 302.  In  multiplication  how  d^^  von  set  down  the  multiplier? 
Where  do  you  begin  to  multiply  ?  Hou  di  you  carry  rroui  une  deiioraina 
tioij.  to  another  ? 


2    X   S^^rzz 

V 

4// 

2X7^=     1 

2' 

2X6=  12 

9'x  S''  — 

6" 

9'x  7^1= 

5' 

3^" 

9^  X  6   =    4 

6' 

Prod.   18 

3' 

K' 

MULTIPLICATION    OF    DUODECIMALS.  317 

Again,  9  inches  or  y^  of  a  foot  multiplied  by  8  twelfths 
of  ^3^  of  a  foot,  will  give  72  twelfths  of  twelfths  of  twelfths 
of  a  square  foot,  which  are  equal  to  six  twelfths  of  twelfths, 
or  to  6''.  Then  9'  X  7'  gives  63  twelfths  of  twelfths  of  a 
square  foot,  equal  to  5^  and  3^^ :  and  9^  X  6  gives  4  square 
feet  and  6^ 

303.  Hence  we  see, 

1st.  T/tat  feet  multiplied  hy  feet  give  square  feet  in  th^ 
pioduct, 

2d.  That  feet  multiplied  hy  inches  give  twelfths  of  square 
feet  in  the  product. 

3d.  That  inches  multiplied  hy  inches  give  twelfths  of  twelfths 
of  square  feet  in  the  product, 

4th.  That  inches  multiplied  hy  seconds  give  twelfths  of 
twelfths  of  twelfths  of  square  feet  in  the  product, 

2.  Multiply  9ft.  4in.  by  9ft.  Sin. 
Beginning  with  the  8  feet,  we 

say  8  times  4  are  32^,  which  is 

€qual  to  2  feet  8^ :  set  down  the 

8^     Then  say  8  times  9  are  72 

and  2  to  carry  are  74  feet :  then 

multiplying  by  3^  we  say,  3  times 

4^  are  1 2^^,  equal  to  1  inch :  set 

down  0  in  the  second's  place :  then  3  times  9  are  27  and  1 

to  carry  make  28^,  equal  to  2ft.  4^.     Therefore  the  entire 

product  is  equal  to  77ft. 

3.  How  many  solid  feet  in  a  stick  of  timber  which  is  25ft. 
6in.  long,  2ft,  7in.  broad,  and  3ft.  Sin.  thick  ? 

4.  Multiply  9ft.  2in.  by  9ft.  6in.  Ans,  

5.  Multiply  S4ft,  lOin.  by  6ft.  Sin.  Ans.  

6.  Multiply  70ft.  9in,  by  12ft.  Sin.  .     Ans.  

7.  How  many  cords  and  cord  feet  in  a  pile  of  wood  24 
feet  long,  4  feet  wide,  and  Sft.  6in.  high  ? 

8.  Multiply  6ft.  9'  hy  8ft.  6'.  Ans.  

Quest. — 30.3.  Repeat  the  four  principles. 


OPERATION* 
9       4" 
8       S' 

74     8^ 
2     4' 

0'' 

77     0' 

0''  A 

318  INVOLUTION. 

9.  How  many  cord  feet  in  a  pile  of  wood  25  feet  long, 
6  feet  wide,  and  5  feet  high  ? 

10.  Multiply  IQft.  9^  by  lift.  \V\  Ans,  

Note. — It  must  be  recollected  that  16  soHd  feet  make  1  cord  foot, 
(Art.  30). 


INVOLUTION. 

304.  If  a  number  be  multiplied  by  itself,  the  product  is 
called  the  second  power,  or  square  of  that  number.  Thus, 
4  X  4  =  16 :  the  number  16  is  the  2d  power  or  square  of  4. 

If  a  number  be  multiplied  by  itself,  and  the  product  arising 
be  again  multiplied  by  the  number,  the  second  product  is 
called  the  3d  power,  or  cube  of  the  number.  Thus,  3x3x3 
=  27  :'  the  number  27  is  the  ^id  power,  or  cube  of  3. 

The  term  power  designates  the  product  arising  from  multi- 
plying a  number  by  itself  a  certain  number  of  times,  and  the 
number  multiplied  is  called  the  root. 

Thus,  in  the  first  example  above,  4  is  the  root,  and  16  the 
square  or  2d  power  of  4. 

In  the  2d  example,  3  is  the  root,  and  27  the  3d  power  or 
cube  of  3.    The  first  power  of  a  number  is  the  number  itself. 

305.  Involution  teaches  the  method  of  finding  the  powers  of 
numbers. 

The  number  which  designates  the  power  to  which  the  root 
is  to  be  raised,  is  called  the  index  or  exponent  of  the  power. 
It  is  generally  written  on  the  right,  and  a  little  above  the  root. 

Quest. — How  many  solid  feet  make  a  cord  foot  ?  304.  If  a  number  be 
multiplied  by  itself  once,  what  is  the  product  called  ?  If  it  be  multiplied  by 
itself  twice,  what  is  the  product  called  ?  What  does  the  term  power  mean  ? 
What  is  the  root  ?  What  is  the  first  power  of  a  number  ?  305.  What  is 
Involution?  What  is  the  nmnber  called  which  designates  the  power? 
Where  is  it  written  ? 


INVOLUTION.  H!9 

Thus,  4^  expresses  the  2d  power  of  4,  or  that  4  is  to  be  mul- 
tiplied by  itself  once  :  hence, 

^  4^  =  4  X  4  =  16. 
For  the  same  reason  3^  denotes  that  3  is  to  be  raised  to 
the  3d  power,  or  cubed :  hence, 

3^  =  3  X  3  X  3  =:  27  :   we  may  therefore  write 

4  rz:         4  the  1st  power  of  4. 
4^  =  4  X  4  z=       16  the  2d   power  of  4. 
4^  =  4x4x4=      64  the  3d   power  of  4. 
4^  =  4x4x4x4=    256  the  4th  power  of  4. 
4*  =  4x4x4x4x4  =  1024  the  5th  power  of  4. 
Sic,  (fee,  &c. 

Hence,  to  raise  a  number  to  any  power, 
Multiply  the  number  continually  hy  itself  as  many  times  less 
1  as  there  are  units  in  the  exponent,  and  the  last  product  will  be 
the  power  sought,  ^ 

EXAMPLES. 

1.  What  is  the  3d  power  of  125  ?  Ans,  

2.  What  is  the  cube  of  7  ?  Ans,  

3.  What  is  the  square  of  60  ?  A71S,  

4.  What  is  the  4th  power  of  5  ?  Ans.  

5.  What  is  the  5th  power  of  18  ?  Ans,  

6.  What  is  the  cube  of  1  ?  Ans,  

7.  What  is  the  square  of  ^  ?  Ans,  

8.  What  is  the  cube  of  .1  ?  Ans.  

9.  What  is  the  cube  of  f  ?  Ans,  

10.  What  is  the  square  of  .01 1  Ans.  

11.  What  is  the  square  of  6.12  ?  Ans.  

12.  What  is  the  6th  power  of  10  1  Ans.  

13.  What  is  the  cube  of  31?  Ans.  

1 4.  What  is  the  4th  power  of  36  ?  Ans.  

15.  What  is  the  cube  of  8733  ?  Ans.  


Quest. — What  is  the  exponent  of  the  square  of  a  number?     Of  the 
cube  ?    Of  the  fourth  power?    How  do  you  raise  a  number  to  any  power  ? 


320  EVOLUTION 


EVOLUTION 

306.  We  have  seen  that  Involution  teaches  how  to  find 
the  povi^er  when  the  root  is  given.  Evolution  is  the  reverse 
of  Involution  :  it  teaches  how  to  find  the  root  when  the  power 
is  known.  The  root  is  that  number  which  being  multiplied 
by  itself  a  certain  number  of  times,  will  produce  the  given 
power. 

The  square  root  of  a  number  is  that  number  which  being 
multiplied  by  itself  once,  will  produce  the  given  number. 

The  cube  root  of  a  number  is  that  number  which  being 
multiplied  by  itself  twice,  will  produce  the  given  number. 

For  example,  6  is  the  square  root  of  36,  because  6x6 
=  36  ;  and  3  is  the  cube  root  of  27,  because  3x3x3=  27. 
The  sign  V  placed  before  a  number  denotes  that  its  square 
root  is  to  be  extracted.  Thus,  y  36  =:  6.  The  sign  V  is 
called  the  radical  sign,  or  the  sign  of  the  square  root. 

When  we  wish  to  express  that  the  cube  root  is  to  be  ex- 
tracted, we  place  the  figure  3  over  the  sign  of  the  square 
root :  thus,  -x/S  =  2  and  V27  =  3,  and  3  is  called  the  index 
of  the  root. 


EXTRACTION  OF  THE  SQUARE  ROOT. 

307.  To  extract  the  square  root  of  a  number,  is  to  find  a 
number  which  being  multiplied  by  itself  once,  will  produce 
the  given  number.     Thus, 

VT=z2;    for    2x2=4; 

V^=z  3;    for    3x3=9. 

Quest.— 306.  What  is  Evolution?  What  does  it  teach ?  What  is  the 
root  of  a  number  ?  What  is  the  square  root  of  a  number  ?  What  is  the 
cube  root  of  a  number?  Make  the  sign  denoting  the  square  root.  How  do 
you  denote  the  cube  root  ?  307.  What  is  required  when  we  wish  to  ex- 
tract the  square  root  of  a  number  ? 


EXTRACTION  OF  THE  SQUARE  ROOT. 


321 


Before  proceeding  to  explain  the  rule  for  extracting  the 
square  root,  let  us  first  see  how  the  squares  of  numbers  are 
formed. 

The  first  ten  numbers  are 

1,     2,     3,     4,     5,     6,     7,     8,     9,     10     Roots. 
1      4      9     16    25    36    49    64    81    100    Squares. 
The  numbers  in  the  second  line  are  the  squares  of  those  in 
the  first ;  and  the  numbers  in  the  first  line   are  the  square 
roots  of  the  corresponding  numbers  of  the  second. 

Now,  it  is  evident  that,  the  square  of  a  number  expressed  hy 
a  single  figure  will  not  contain  any  figure  of  a  higher  order  than 
tens ;  and  also,  that  if  a  numher  contains  three  figures,  its  root 
must  contain  tens  and  units 

The  numbers  1,  4,  9,  &c.,  of  the  second  line,  are  called 
perfect  squares,  because  they  have  exact  roots. 

Let  us  now  see  how  the  square  of  any  number  may  be 
formed,  say  the  number  36.  This  number  is  made  up  of  3 
tens  or  30,  and  6  units 

Let  the  line  AB  represent 
the  3  tens  or  30,  and  BC  the 
six  units. 

Let  AD  be  a  square  on 
AC,  and  AE  a  square  on  the 
tens  line  AB. 

Then  ED  will  be  a  square 
on  the  unit  line  6,  and  the 
rectangle  EF  will  be  the 
product  of  HE  which  is  equal 
to  the  tens  line,  by  IE  which 
is  equal  to  the  unit  line.  Also, 
the  rectangle  BK  will  be  the  product  of  EB  which  is  equal 
to  the  tens  line,  by  the  unit  line  BC.     But  the  whole  square 


Quest. — What  is  the  greatest  square  of  a  single  figure  ?  What  is  the 
highest  order  of  units  that  can  be  derived  from  the  square  of  a  single  fig- 
ure ?  How  many  perfect  squares  are  there  among  the  numbers  that  are 
less  than  one  hmidred? 

14* 


322 


EVOLUTiON. 


12  96(36 
9 
66)396 
396 


30 


on  AC  is  made  up  of  the  square  AE,  the  two  rectangles  FE 
and  EC,  and  the  square  ED.     Hence, 

The  square  of  two  figures  is  equal  to  the  square  of  the  tens, 
plus  twice  the  product  of  the  tens  by  the  units,  plus  the  square 
of  the  units. 

Let  it  now  be  required  to  extract  the  square  root  of  1296. 

Since  the  number  contains  more  than  two  places,  its  root 
will  contain  tens  and  units.  But  as  the  square  of  one  ten  is 
one  hundred,  it  follows  that  the  ten's  place  of  the  required 
root  must  be  found  in  the  figures  on  the  left  of  96.  Hence, 
we  point  off  the  number  into  periods  of  two 
figures  each. 

We  next  find  the  greatest  square  con- 
tained in  12,  which  is  3  tens  or  30.  We 
then  square  3  tens  which  gives  9  hundred, 
and  then  place  9  under  the  hundred's  place,  axid  subtract. 

This  takes  away  the  square 
AE  and  leaves  the  two  rect- 
angles FE  and  BK,  together 
with  the  square  ED  on  the 
unit  line. 

Now,  since  tens  multiplied 
by  units  will  give  at  least 
tens  in  the  product,  it  follows 
that  the  area  of  the  two  rect- 
angles FE  and  EC  must  be 
expressed  by  the  figures  of 
the  given  number  at  the  left 
of  the  unit's  place  6,  which 
figures  may  also  express  a  part  of  the  square  ED. 

If,  then,  we  divide  the  figures  39,  at  the  left  of  6,  by  twice 
the  tens,  that  is,  by  twice  AB  or  BE,  the  quotient  will  be 
BC  or  EK,  the  unit  of  the  root. 

Quest. — ^What  is  the  square  of  a  number  expressed  by  two  fignres  equal 
to  ?  In  what  places  of  figures  will  the  square  of  the  tens  be  found  ?  In 
what  places  will  the  product  of  the  tens  by  the  units  be  found  ? 


30 

6 

6 

6 

180 

36 

30         E 

900+180+180+36=  1296. 

^ 

,30 

30 

30 

6 

900 

180 

EXTRACTION  OF  THE  SQUARE  ROOT.       323 

Then,  placing  BC  or  6,  in  the  root,  and  also  in  the  divisor 
and  then  multiplying  the  whole  divisor  66  by  6,  we  obtain 
for  a  product  the  two  rectangles  FE  and  CE,  together  with 
the  square  ED. 

Hence,  the  square  root  of  1296  is  36;  or,  in  other  words,  36 
is  the  side  of  a  square  whose  area  is  1296. 


308.  To  extract  the  square  root  of  a  whole  number, 

I.  Point  off  the  given  number  into  periods  of  two  figures 
each,  counted  from  the  right,  by  setting  a  dot  over  the  place  of 
units,  another  over  the  place  of  hundreds,  and  so  on, 

II.  Find  the  greatest  square  in  the  first  period  on  the  left, 
and  place  its  root  on  the  right  after  the  manner  of  a  quotient  in 
division.  Subtract  the  square  of  the  root  from  the  first  period, 
and  to  the  remainder  bring  down  the  second  period  for  a  divi- 
dend, 

III.  Double  the  root  already  found  and  place  it  on  the  left 
for  a  divisor.  Seek  how  many  times  the  divisor  is  contained 
in  the  dividend,  exclusive  of  the  right  hand  figure,  and  place  the 
figure  in  the  root  and  also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor,  thus  augmented,  by  the  last  figure 
of  the  root,  and  subtract  the  product  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  dividend. 
But  if  the  product  should  exceed  the  dividend,  diminish  the  last 
figure  of  the  root. 

V.  Double  the  whole  root  already  found,  for  a  new  divisor, 
and  continue  the  operation  as  before,  until  all  the  periods  are 
brought  down. 

EXAMPLES. 

1.  What  is  the  square  root  of  263169  ? 

Quest. — 308.  What  is  the  first  step  in  extracting  the  square  root  of 
numbers?  What  the  second ?  What  the  third?  What  the  fourth ?  What 
the  fifth  ?     Give  the  entire  rule. 


324 


EVOLUTION. 


26  31   69(513 
25 

101)131 
101 

1023)3069 
3069 


We  first  place  a  dot  over  the  9,  operation. 

making  the  right  hand  period  69. 
We  then  put  a  dot  over  the  1  and 
also  over  the  6,  making  three  pe- 
riods. 

The  greatest  perfect  square  in 
26,  is  25,  the  root  of  M^hich  is  5. 
Placing  5  in  the  root,  subtracting  its 
square  from  26,  and  bringing  down  the  next  period  31,  we 
have  131  for  a  dividend,  and  by  doubling  the  root  we  have 
10  for  a  divisor.  Now  10  is  contained  in  13,  1  time.  Place 
1  both  in  the  root  and  in  the  divisor:  then  multiply  101  by 
1  ;  subtract  the  product  and  bring  down  the  next  period. 

We  must  now  double  the  whole  root  51  for  a  new  divisor, 
or  we  may  take  the  first  divisor  after  having  doubled  the  last 
figure  1  ;  then  dividing  we  obtain  3,  the  third  figure  of  the  root. 

309.  Note  1. — There  will  be  as  many  figures  in  the  root  as 
there  are  periods  in  the  given  number. 

Note  2. — If  the  given  number  has  not  an  exact  root,  there  will 
be  a  remainder  after  all  the  periods  are  brought  down,  in  which 
case  ciphers  may  be  annexed,  forming  new  periods,  each  of  which 
will  give  one  decimal  place  in  the  root. 

2.  What  is  the  square  root  of  36729  1 


OPERATION. 

3  67  29(191.64-f-. 
1 

29)267 

In   this    example   there 

261 

are    two  places    of   deci- 

381)629 

mals,  which  give  two  pla- 

381 

ces  of  decimal  in  the  root. 

3826)24800 

22956 

38324)184400 

153296 

31104  Rem. 

Quest.— 309.  How  many  figxu-es  will  there  be  in  the  root?     If  the 
given  number  has  not  an  exact  root,  what  may  be  done  ? 


EXTRACTION    OF    THE    SQUARE    ROOT.  325 

3.  What  is  the  square  root  of  213444  ?  Ans.  

4.  What  is  the  square  root  of  2268741  ?  Ans.  

5.  What  is  the  square  root  of  15193592  ?  Ans.  

6.  What  is  the  square  root  of  36372961  ?         Ans.  

7.  What  is  the  square  root  of  22071204  ?         Ans.  


310.  To  extract  the  square  root  of  a  decimal  fraction, 

I.  Annex  one  cipher,  if  necessary/,  so  that  the  number  of  deci- 
mal places  shall  be  even. 

II.  Point  off  the  decimals  into  periods  of  two  figures  each, 
by  putting  a  point  over  the  place  of  hundredths,  a  second,  over 
the  place  often  thousandths,  dSfC.  :  then  extract  the  root  as  in 
whole  numbers,  recollecting  that  the  number  of  decimal  places  in 
the  root  will  be  equal  to  the  number  of  periods  in  the  given 
decimal, 

EXAMPLES. 

1.  What  is  the  square  root  of  .5  ? 


We  first  annex  one  cipher  to 
make  even  decimal  places.  We 
then  extract  the  root  of  the  first 
period,  to  which  we  annex  ci- 
phers, forming  new  periods. 


OPERATION. 

.50(.707  + 
49 

140)100        ^ 
000 


1407)10000 
9849 


151   Rem. 


Note. — When  there  is  a  decimal  and  a  whole  number  joined  to- 
gether the  same  rule  will  apply. 

2.  What  is  the  square  root  of  3271.4207  ?  A71S.  

3.  What  is  the  square  root  of  4795.25731  ?  Ans.  

4.  What  is  the  square  root  of  4.372594  ?  Ans.  

5.  What  is  the  square  root  of  .00032754  ?  Ans.  

Quest. — 310.  How  do  you  extract  the  square  root  of  a  decimal  frac- 
tion ?  When  there  is  a  decimal  and  a  whole  number  joined  together,  will 
the  same  rule  apply  ? 


ns. 


326  EVOLUTION. 

6.  What  is  the  square  root  of  .00103041  ?  A..o. 

7.  What  is  the  square  root  of  4.426816  1  Ans, 

8.  What  is  the  square  root  of  47.692836  ?  Ans 


311.  To  extract  the  square  root  of  a  vulgar  fraction, 

I.  Reduce  mixed  numbers  to  improper  fractions^  and  com- 
pound fractions  to  simple  ones,  and  then  reduce  the  fraction  to 
its  lowest  terms, 

II.  Extract  the  square  root  of  the  numerator  and  denoinina- 
tor  separately,  if  they  have  exact  roots ;  hut  when  they  have 
not,  reduce  the  fraction  to  a  decimal  and  extract  the  root  as  in 
Case  II, 

EXAMPLES. 

1 .  What  is  the  square  root  of  W  of  ^^y  of  4i  ? 

2.  What  is  the  square  root  of  ||f|?  Ans.  

3.  What  is  the  square  root  of  y^2_l6_  ?  ^^^^  

4.  What  is  the  square  root  of  |-^-|?  Ans,  

5.  What  is  the  square  root  of  \W  ?  Ans,  

6.  What  is  the  square  root  of  ||-|  ?  Ans, 


EXTRACTION  OF  THE  CUBE  ROOT. 

312.  To  extract  the  cube  root  of  a  number  is  to  find  a 
second  number  which  being  multiplied  into  itself  twice,  shall 
produce  the  given  number. 

Thus,  2  is  the  cube  root  of  8 ;  for,  2  X  2  x  2  =z  8 :  and  3 
is  the  cube  root  of  27 ;  for,  3  X  3  X  3  =  27. 
Roots  1,     2,      3,      4,'       5,        6,         7,         8,         9. 

Cubes  1       8      27     64      125     216      343      512       729 

Quest. — 311.  How  do  you  extract  the  square  root  of  a  vulgar  frac- 
tion ? 


EXTRACTION  OF  THE  CUBE  ROOT.        327 

From  which  we  see,  that  the  cube  of  units  will  not  give 
a  higher  order  than  hundreds.  We  may  also  remark,  that 
the  cube  of  one  ten  or  10,  is  1000  :  and  the  cube  of  9  tens 
or  90,  729000 ;  and  hence,  the  cube  of  tens  will  not  give  a 
lower  denomination  than  thousands,  nor  a  higher  denomination 
than  hundreds  of  thousands.  Hence  also,  if  a  number  contains 
more  than  three  figures  its  cube  root  will  contain  more  than 
one  ;  if  the  number  contains  more  than  six  figures  the  root 
will  contain  more  than  two ;  and^  so  on,  every  three  figures 
from  the  right  giving  one  additional  place  in  the  root,  and  the 
figures  which  remain  at  the  left  hand,  although  less  than 
three,  will  also  give  one  place  in  the  root. 

Let  us  now  see  how  the  cube  of  any  number,  as  16,  is 
formed.  Sixteen  is  composed  of  1  ten  and  6  units,  and  may 
be  written  10  +  6.  Now  to  find  the  cube  of  16  or  of  10  -f-  6, 
we  must  multiply  the  number  by  itself  twice. 

To  do  this  we  place  the  numbers  thus  10  +      6 

10  +      6 

Product  by  the  units 60  +    36 

Product  by  the  tens     -       -       -       -100+60 

Square  of  16, 100  +    120  +    36 

Multiply  again  by  16 10+       6 

Product  by  the  units  -  -  -  -  600  +  720  +  216 
Product  by  the  tens     -       -    1000  +  1200  +    360 

Cube  of  16      -         -       -    1000  +  180^+  1080  +  216 

1 .  By  examining  the  composition  of  this  number  it  will  be 
[found  that  the  first  part  1000  is  the  cube  of  the  tens  ;  that  is, 

10  X  10  X  10  =  1000. 

2.  The  second  part  1 800  is  equal  to  three  times  the  square 
of  the  tens  multiplied  by  the  units  ;  that  is, 

3  X  (10)'  X6  =  3xl00x6=:  1800. 

,  3.  The  third  part  1080  is  equal  to  three  times  the  square 
of  the  units  multiplied  by  the  tens  ;  that  is, 

3  X  6'  X  10  =  3  X  36  X  10  =  1080. 


OPERATION. 


4  096(16 

1 

1^  X  3  =:  3)3  0     (9-8-7-6 
16'  =:  4  096. 


328  EVOLUTION. 

4.  The  fourth  part  is  equal  to  the  cube  of  the  units  ;  that  is, 
6'  =  6  X  6  X  6  =  216. 

Let  it  now  be  required  to  extract  the  cube  root  of  the  num- 
ber 4096. 

Since  the  number  con- 
tains more  than  three  fig- 
ures, we  know  that  the  root 
will  contain  at  Ifeast  units 
and  tens.  * 

Separating  the  three  right 
hand  figures  from  the  4,  we  know  that  the  cube  of  the  tens 
will  be  found  in  the  4.     Now,  1  is  the  greatest  cube  in  4. 

Hence,  we  place  the  roo^  1  on  the  right,  and  this  is  the 
tens  of  the  required  root.  We  then  cube  1  and  subtract  the 
result  from  4,  and  to  the  remainder  we  bring  down  the  first 
figure  0  of  the  next  period. 

Now,  we  have  seen  that  the  second  part  of  the  cube  of  16, 
viz.,  1800,  being  three  times  the  square  of  the  tens  multiplied 
by  the  units,  will  have  no  significant  figure  of  a  less  denomi- 
nation than  hundreds,  and  consequently  will  make  up  a  part 
of  the  30  hundreds  above.  But  this  30  hundreds  also  con- 
tains all  the  hundreds  which  come  from  the  3d  and  4th  parts 
of  the  cube  of  16.  If  this  were  not  the  case,  the  30  hundreds 
divided  by  three  times  the  square  of  the  tens  would  give  the 
unit  figure  exactly. 

Forming  a  divisor  of  three  times  the  square  of  the  tens,  we 
find  the  quotient  to  be  ten  ;  but  this  we  know  to  be  too  large. 
Placing  9  in  the  root  and  cubing  19,  we  find  the  result  to  be 
6859.  Then  trying  8  we  find  the  cube  of  18  still  too  large ; 
but  when  we  take  6  we  find  the  exact  i\umber.  Hence,  the 
cube  root  of  4096  is  16. 

CASE    I. 

313.  To  extract  the  cube  root  of  a  whole  number, 
I.  Point  off  the  given  number  into  periods  of  three  figures 
each,  by  placing  a  dot  oiier  the  place  of  units ,  a  second  over  the 


EXTRACTION  OF  THE  CUBE  ROOT.        329 

place  of  thousands,  and  so  on  to  the  left :  the  left  hand  period 
will  often  contain  less  than  three  places  of  figures. 

II.  Seek  the  greatest  cube  in  the  first  period,  and  set  its  root 
on  the  right  after  the  manner  of  a  quotient  in  division.  Sub- 
tract the  cube  of  this  figure  from  the  first  period,  and  to  the 
remainder  bring  down  the  first  figure  of  the  next  period,  and 
call  this  number  the  dividend, 

III.  Take  three  times  the  square  of  the  root  just  found  for  a 
divisor  and  see  how  often  it  is  contained  in  the  dividend,  and 
place  the  quotient  for  a  second  figure  of  the  root.  Then  cube 
the  figures  of  the  root  thus  found,  and  if  their  cube  be  greater 
than  the  first  two  periods  of  the  given  number,  diminish  the  last 
figure,  but  if  it  be  less,  subtract  it  from  the  first  two  periods, 
and  to  the  remainder  bring  down  the  first  figure  of  the  next 
period,  for  a  new  dividend. 

IV.  Take  three  times  the  square  of  the  whole  root  for  a  new 
divisor,  and  seek  how  often  it  is  contained  in  the  new  dividend : 
the  quotient  will  be  the  third  figure  of  the  root.  Cube  the  whole 
root  and  subtract  the  result  from  the  first  three  periods  of  the 
given  number,  and  proceed  in  a  similar  way  for  all  the  periods. 

EXAMPLES. 

1.  What  is  the  cube  root  of  99252847  ? 

99  252  847(463 
4^  =:  64 
4^  X  3  =  48")352      dividend 
First  two  periods     -     -     -     -     99  252 . 
(46y  =  46  X  46  X  46  z=  97  336 


3  X  (46)2  =  G348  )     19168  2d  dividend 
The  first  three  periods       -     -     99  252  847 
(463)'  =  99  252  847 

Ans.  463, 

Quest. — 312.  What  is  required  when  we  are  to  extract  the  cube  root  of 
a  number?    313    How  do  you  extract  the  cube  root  of  a  whole  number? 


330  EVOLUTION. 

2.  What  is  the  cube  root  of  389017?  Ans.  . 

3.  What  is  the  cube'root  of  5735339  ?  Ans,  

4.  What  is  the  cube  root  of  32461759  ?  Ans.  

5.  What  is  the  cube  root  of  84604519  ?  Ans,  

6.  What  is  the  cube  root  of  259694072  ?  Ans,  

7.  What  is  the  cube  root  of  48228544  ?  Ans,  

8.  What  is  the  cube  root  of  27054036008  ?  Ans,  

CASE    II. 

314.  To  extract  the  cube  root  of  a  decimal  fraction, 
Annex  ciphers  to  the  decimals,  if  necessary,  so  that  it  shaK 
consist  of  3,  6,  9,  <^c.,  places.  Then  put  the  first  point  over 
the  place  of  thousandths,  the  second  over  the  place  of  millionths, 
and  so  on  over  every  third  place  to  the  right ;  after  which  ex- 
tract the  root  as  in  whole  numbers. 

Note  1. — There  will  be  as  many  decimal  places  in  the  root  as 
there  are  periods  in  the  given  number. 

Note  2. — The  same  rule  applies  when  the  given  number  is  com- 
posed of  a  whole  number  and  a  decimal. 

Note  3. — If  in  extracting  the  root  of  a  number  there  is  a  re 
mainder  after  all  the  periods  have  been  brought  down,  periods  of 
ciphers  may  be  annexed  by  considering  them  as  decimals. 

EXAMPLES. 

1.  What  is  the  cube  root  of  .127464  ?  Ans,  

2.  What  is  the  cube  root  of  .870983875  ?  Ans,  

3.  What  is  the  cube  root  of  12.977875  ?  Ans,  

4.  What  is  the  cube  root  of  75.1089429  ?  Ans,  

5.  What  is  the  cube  root  of  .353393243  ?  Ans,  

6.  What  is  the  cube  root  of  3.408862625  ?  Ans,  

7.  What  is  the  cube  root  of  27.708101576  ?  Ans.  

Quest. — 314.  How  do  you  extract  the  cube  root  of  a  decimal  fraction' 
How  many  decimal  places  will  there  be  in  the  root?  Will  the  same  rule 
apply  when  there  is  a  whole  number  and  a  decimal  ?  In  extractmg  the 
root  if  there  is  a  remainder,  what  may  be  done  ? 


ARITHMETICAL    PROGRESSION.  331 

CASE    III. 

315.  To  extract  the  cube  root  of  a  vulgar  fraction, 

I.  Reduce  compound  fractions  to  simple  ones,  mixed  numbers 
to  improper  fractions,  and  then  reduce  the  fraction  to  its  lowest 
terms. 

II.  Then  extract  the  cube  root  of  the  numerator  and  denomi- 
nator separately,  if  they  have  exact  roots  ;  but  if  either  of  them 
has  not  an  exact  root,  reduce  the  fraction  to  a  decimal,  and  ex- 
tract the  root  as  in  the  last  Case. 

EXAMPLES. 

1.  What  is  the  cube  root  of  ||f  ?  Ans.  

2.  What  is  the  cube  root  of  12if  ^  Ans.  

3.  What  is  the  cube  root  of  31^^3  ?  Ans.  

4.  What  is  the  cube  root  of  y^o^  ?  Ans.  

5.  What  is  the  cube  root  of  ^  1  Ans.  

6.  What  is  the  cube  root  of  |  ?  Ans.  

7.  What  is  the  cube  root  of  f  ?  Ans.  


ARITHMETICAL  PROGRESSION. 

316.  If  we  take  any  number,  as  2,  we  can,  by  the  con 
tinned  addition  of  any  other  number,  as  3,  form  a  series  of 
numbers :  thus, 

2,    5,    8,     11,     14,     17,    20,    23,    &c., 
in  which  each  number  is  formed  by  the  addition  of  3  to  the 
preceding  number. 

This  series  of  numbers  may  also  be  formed  by  subtracting 
3  continually  from  the  larger  number :  thus, 

23,    .20,     17,     14,     11,    8,    5,    2. 

A  series  of  numbers  formed  in  either  way  is  called  an 
Arithmetical  Series,  or  an  Arithmetical  Progression ;  and  the 

Quest. — 315.  How  do  you  extract  the  "cube  root  of  a  vulgar  fractiou? 
316.  How  do  you  form  an  Arithmetical  Series'? 


332  ARITHMETICAL    PROGRESSION. 

number  which  is  added  or  subtracted  is  called  the  common 
difference. 

When  the  series  is  formed  by  the  continued  addition  of  the 
common  difference,  it  is  called  an  ascending  series  ;  and 
when  it  .is  formed  by  the  subtraction  of  the  common  differ 
ence,  it  is  called  a  descending  series ;  thus, 

2,     5,     8,    11,    14,  17,  20,  23,  is  an  ascending  series 
23,  20,  17,    14,    11,     8,     5,     2,  is  a  descending  series 

The  several  numbers  are  called  terms  of  the  progressia  . 
the  first  and  last  terms  are  called  the  extremes,  and  the  in*  - 
mediate  terms  are  called  the  means, 

317.  In  every  arithmetical  progression  there  are  five  th  gs 
which  are  considered,  any  three  of  which  being  give^^  ox 
known,  the  remaining  two  can  be  determined.     They  aie 

1  st,  the  first  term ; 

2d,   the  last  term ; 

3d,   the  common  difference  ; 

4th,  the  number  of  terms  ; 

5th,  the  sum  of  all  the  terms. 

318.  By  considering  the  manner  in  which  tb  ascendmg 
progression  is  formed,  we  see  that  the  2d  term  ie  obtained  by 
adding  the  common  difference  to  the  1st  term  ;  the  3d,  by 
adding  the  common  difference  to' the  2d;  the  4thj  by  adding 
the  common  difference  to  the  3d,  and  so  on ;  the  number  of 
additions  being  1  less  than  the  number  of  terms  fouT\d, 

But  instead  of  making  the  additions,  we  may  multiply  the 
common  difference  by  the  number  of  additions,  that  is,  by  1 
less  than  the  number  of  terms,  and  add  the  first  term  to  the 
product.     Hence,  we  have 

Quest. — What  is  the  common  difference  ?  What  is  an  ascending  series  ? 
What  a  descending  series  ?  What  are  the  several  numbers  called  ?  What 
are  the  first  and  last  terms  called  ?  What  are  the  intermediate  terms  called  ? 
317.  In  every  eirithmetical  progression  how  many  things  are  considered? 
What  are  they?  318.  How  do  you  find  the  last  term  when  the  first  term 
and  conamon  difference  are  known  ? 


ARITHMETICAL    PJIOGRESSION.  333 

CASE    I. 

Having  given  the  first  term,  the  common  difference,  and 
the  number  of  terms,  to  find  the  last  term. 

Multiply  the  common  difference  hy  1  less  than  the  number  of 
terms,  and  to  the  product  add  the  first  term. 

EXAMPLES. 

1.  The  first  term  is  3,  the  common  difference  2,  and  the 
number  of  terms  1 9  :  what  is  the  last  term  ? 

OPERATION. 


We  multiply  the  number 
of  terms  less  1,  by  the  com- 
mon difference  2,  and  then 
add  the  first  term. 


18  number  of  terms  less  1. 

2  common  difference 
"36" 

3  1st  term. 
39  last  term. 


2.  A  man  bought  50  yards  of  cloth ;  he  was  to  pay  6  cents 
for  the  first  yard,  9  cents  for  the  2d,  12  cents  for  the  3d,  and 
so  on  increasing  by  the  common  difference  3  :  how  much 
did  he  pay  for  the  last  yard  ? 

3.  A  man  puts  out  $100  at  simple  interest,  at  7  per  cent ; 
at  the  end  of  the  first  year  it  will  have  increased  to  $107,  at 
the  end  of  the  2d  year  to  $1 14,  and  so  on,  increasing  $7  each 
year  :  what  will  be  the  amount  at  the  end  of  16  years  ? 

319.  Since  the  last  term  of  an  arithmetical  progression  is 
equal  to  the  first  term  added  to  the  product  of  the  common 
difference  by  1  less  than  the  number  of  terms,  it  follows,  that 
the  difference  of  the  extremes  will  be  equal  to  this  product, 
and  that  the  common  difference  will  be  equal  to  this  product 
divided  by  1  less  than  the  number  of  terms.  Hence,  we 
have 

CASE    II. 

Having  given  the  two  extremes  and  the  number  of  terms 

of  an  arithmetical  progression,  to  find  the  common  difference. 

Subtract  the  less  extreme  from  the  greater  and  divide  the  re- 

Quest. — 319.  How  do  you  find  the  common  diiFerence,  when  you  know 
the  two  extremes  and  number  of  terms  ? 


334  ARITHMETICAL    PROGRESSION. 

mainder  by  1  less  than  the  number  of  terms :  the  quotient  will 
he  the  common  difference. 

EXAMPLES. 

1.  The  extremes  are  4  and  104,  and  the  number  of  terras 
.  26  :  what  is  the  common  difference  ? 

We  subtract  the  less  ex- 
treme from  the  greater  and 
divide  the  difference  by  one 
less  than  the  number  of 
terms. 


OPERATION. 

104 
4 


26-  1  =1  25)100(4 
100 


2.  A  man  has  8  sons,  the  youngest  is  4  years  old  and  the 
eldest  32,  their  ages  increase  in  arithmetical  progression : 
what  is  the  common  difference  of  their  ages  ? 

3.  A  man  is  to  travel  from  New  York  to  a  certain  place  in 
1 2  days  ;  to  go  3  miles  the  first  day,  increasing  every  day 
by  the  same  number  of  miles ;  so  that  the  last  day's  journey 
may  be  58  miles  :  required  the  daily  increase. 

320.  If  we  take  any  arithmetical  series,  as 
3     5     7     9   11   13   15   17  19,   &c. 

19   17    15    lo    11      9      7      O      o      by  reversing  the  order  of 

22  22  22  22  22  22  22  22  22  |  the  terms. 

Here  we  see  that  the  sum  of  the  terms  of  these  two  series 
is  equal  to  22,  the  sum  of  the  extremes,  multiplied  by  the 
number  of  terms  ;  and  consequently,  the  sum  of  either  series 
is  equal  to  the  sum  of  the  two  extremes  multiplied  by  half  the 
number  of  terms  ;  hence,  we  have 

CASE    III. 

To  find  the  sum  of  all  the  terms  of  an  arithmetical  pro- 
gression, 

Add  the  extremes  together  and  multiply  their  sum  by  half  the 
number  of  terms  :  the  product  will  be  sum  of  the  series. 

EXAMPLES. 

1.  The  extremes  are  2  and  100,  and  the  number  of  termi 
22  :  what  is  the  sum  of  the  series  ? 

Quest. — 320.  How  do  you  find  the  sum  of  an  arithmetical  series  ? 


ARITHMETICAL    PROGRESSION.  335 


We  first  add  together 
the  two  extremes,  and 
then  multiply  by  half  the 
number  of  terms. 


OPERATION. 


2  1st  term 
100  last  term 


102  sum  of  extremes 
1 1  half  the  number  of  terms 


1122  sum  of  series. 


2.  How  many  times  does  the  hammer  of  a  clock  strike  in 
12  hours  1 

3.  The  first  term  of  a  series  is  2,  the  common  difference 
4,  and  the  number  of  terms  9  :  what  is  the  last  term  and 
sum  of  the  series  ? 

4.  If  100  eggs  are  placed  in  a  right  line,  exactly  one  yard 
from  each  other,  and  the  first  one  yard  from  a  basket,  what 
distance  will  a  man  travel  who  gathers  them  up  singly,  and 
places  them  in  the  basket  ? 

GENERAL    EXAMPLES. 

1.  What  is  the  18th  term  of  an  arithmetical  progression 
of  which  the  first  term  is  4  and  the  common  difference  5  ? 

2.  The  18th  term  of  an  arithmetical  progression  is  89  and 
the  common  difference  5  :  what  is  the  first  term  1 

3.  A  flight  of  stairs  has  18 -steps  ;  the  first  ascends  but  12 
inches  in  a  vertical  line,  and  each  of  the  others  18  :  what  is 
the  entire  ascent  in  a  vertical  line  ? 

4.  A  debtor  has  18  creditors  ;  he  owes  to  the  largest  cre- 
ditor 89  dollars,  and  5  dollars  less  to  each  of  the  others  in 
succession :  how  much  does  he  owe  to  the  least  ? 

5.  A  person  travelled  from  Boston  to  a  certain  place  in  8 
days  ;  he  travelled  2  miles  the  first  day,  and  every  succeed- 
ing day  he  travelled  farther  than  he  did  the  preceding  by  an 
equal  number  of  miles :  the  last  day  he  travelled  23  miles : 
how  much  did  he  travel  each  day,  and  how  much  in  all  ? 

6.  The  number  of  terms  is  22,  the  common  difference  5, 
and  the  sum  of  the  terms  1221  :  what  is  the  least  term  1 

7.  A  man  is  to  receive  $3000  in  12  payments,  each  suc- 
ceeding payment  to  exceed  the  previous  by  $4 :  what  will 

he  last  payment  be  ? 


336  GEOMETRICAL   PROGRESSION. 


GEOMETRICAL  PROGRESSION 

321.  If  we  take  any  number,  as  3,  and  multiply  it  con* 
tinually  by  any  other  number,  as  2,  we  form  a  series  of  num- 
bers :  thus, 

3     6     12     24     48     96     192,  &c., 
in  which  each  number  is  formed  by  multiplying  the  number 
before  it  by  2. 

This  series  may  also  be  formed  by  dividing  continually  the 
lars^est  number  192  by  2.     Thus, 

192     96     48     24     12     6     3. 

A  series  formed  in  either  way,  is  called  a  Geometrical 
Series,  or  a  Geometrical  Progression,  and  the  number  by 
which  we  continually  multiply  or  divide,  is  called  the  com' 
mon  ratio.  , 

When  the  series  is  formed  by  multiplying  continually  by 
the  common  ratio,  it  is  called  an  ascending  series ;  and  when 
it  is  formed  by  dividing  continually  by  the  common  ratio,  it  is 
called  a  descending  series.     Thus, 

3       6     12    24    48    96     192  is  an  ascending  series. 

192    96    48    24     12      6        3     is  a  descending  series. 

The  several  numbers  are  called  terms  of  the  progression. 

The  first  and  last  terms  are  called  the  extremes,  and  the 
intermediate  terms  are  called  the  means. 

322.  In  every  Geometrical,  as  well  as  in  every  Arithmeti- 
cal Progression,  there  are  five  things  which  are  considered, 
any  three  of  which  being  given  or  known,  the  remaining  two 
can  be  determined.     They  are. 

Quest. — 321.  How  do  you  form  a  Geometrical  Progression?  What  is 
the  common  ratio  ?  What  is  an  ascending  series  ?  What  is  a  descending 
series  ?  What  are  the  several  numbers  called?  What  are  the  first  and  last 
terms  called?  What  are  the  intermediate  terms  called?  322.  Ir  every 
geometrical  progression,  how  many  things  are  considered?  What  are 
they? 


GEOMETRICAL    PROGRESSION.  337 

1st,  the  first  term, 
2d,   the  last  term, 
3d,    the  common  ratio, 
4th,  the  number  of  terms, 
5th,  the  sum  of  all  the  terms. 
By  considering  the  manner  in  which  the  ascending  pro- 
gression is  formed,  we  see  that  the  second  term  is  obtained 
by  multiplying  the  first  term  by  the  common  ratio ;  the  3d 
term  by  multiplying  this  product  by  the  common  ratio,  and  so 
on,  the  number  of  multiplications  being  one  less  than  the 
number  of  terms.     Thus, 

3  =  3     1st  term, 
3x2  =  6     2d   term, 
3  X  2  X  2  =  12  3d   term, 
3  X  2  X  2  X  2  =  24  4th  term,  &c.  for  the  other  terms. 
But  2  X  2  =  2^  2  X  2  X  2  =  2^  and   2  X  2  X  2  X  2  =  2^ 
Therefore,  any  term  of  the  progression  is  equal  to  the  first 
term  multiplied  by  the  ratio  raised  to  a  power  1  less  than  the 
number  of  the  term. 

CASE    I. 

Having  given  the  first  term,  the  common  ratio,  and  th© 
number  of  terms,  to  find  the  last  term. 

Raise  the  ratio  to  a  power  whose  exponent  is  one  less  than 
the  number  of  terins^  and  then  multiply  the  power  by  the  first 
term :  the  product  will  be  the  last  term. 

EXAMPLES. 

1.  The  first  term  is  3  and  the  ratio  2  :  what  is  the  6th 
term  ? 

2X2X2X2X2=  2*  =  32 

3  1st  term. 
Ans,  96 

Quest. — How  many  must  be  known^J^efore  the  remaining  ones  can  be 
iound?    What  is  any  term  equal  to  ?     How  do  you  find  the  last  term? 

15 


388  GEOMETRICAL    PROGRESSION. 

2.  A  man  purchased  12  pears:  he  was  to  pay  1  farthing 
for  the  1st,  2  farthings  for  the  2d,  4  for  the  3d,  and  so  on 
doubling  each  time  :  what  did  he  pay  for  the  last  1 

3.  A  gentleman  dying  left  nine  sons,  and  bequeathed  his 
estate  in  the  following  manner :  to  his  executors  £50 ;  his 
youngest  son  to  have  twice  as  much  as  the  executors,  and 
each  son  to  have  double  the  amount  of  the  son  next  younger : 
what  was  the  eldest  son's  portion  1 

4.  A  man  bought  12  yards  of  cloth,  giving  3  cents  for  the 
1st  yard,  6  for  the  2d,  12  for  the  3d,  &c. :  what  did  he  pay 
for  the  last  yard  1 

CASE    11. 

323.  Having  given  the  ratio  and  the  tw§  extremes  to  find 
the  sum  of  the  series. 

Subtract  the  less  extreme  from  the  greater,  divide  the  remain- 
der by  1  less  than  the  ratio,  and  to  the  quotient  add  the  greater 
extreme :  the  sum  will  be  the  sum  of  the  series. 

EXAMPLES. 

1.  The  first  term  is  3,  the  ratio  2,  and  last  term  192  :  what 
is  the  sum  of  the  series  ? 

192  —  3  =  189  difference  of  the  extremes, 
2 —  1  =  1)189(189;  then  189 +  192  =  381  Ans, 

2.  A  gentleman  married  his  daughter  on  New  Year's  day, 
and  gave  her  husband  Is.  towards  her  portion,  and  was  to 
double  it  on  the  first  day  of  every  month  during  the  year : 
what  was  her  portion  ? 

3.  A  man  bought  10  bushels  of  wheat  on  the  condition 
that  he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  2d, 
9  for  the  3d,  and  so  on  to  the  last :  what  did  he  pay  for  the 
last  bushel  and  for  the  10  bushels  1 

4.  A  man  has  six  children;  to  the  1st  he  gives  $150,  to 
the  2d  $300,  to  the  3d  $600,  and  so  on,  to  each  twice  as 
much  as  the  last :  how  much  did  the  eldest  receive,  and  what 
was  the  amount  received  by  them  all  ? 

^d  111  >  .■,,.iij  : [ : 

V  III-.  >  fluKBT. — ^223.  How  do  you  find  the  sum  of  the  series  1 


MENSURATION.  339 


MENSURATIOK 

324.  Mensuration  is  the  process  of  determining  the  contents 
of  geometrical  figures,  and  is  divided  into  two  parts,  the  men- 
'snration  of  surfaces  and  the  mensuration  of  solids. 

MENSURATION     OF     SURFACES. 

325.  Surfaces  have  length  and  breadth.  They  are  mea- 
sured by  means  of  a  square,  which  is  called  the  unit  of  surface, 

A    square    is    the    space    included    between  i  Foot, 

four  equal   lines,  drawn    perpendicular  to    each     ^ 
other.    ^Each  line  is  called  a  side  of  the  square.     ^ 
If  each  side  be  one  foot,   the  figure  is  called  a 
square  foot 

If  the  sides  of  a  square  be  each  four  feet,  the  square  will 
contain  sixteen  square  feet.  For,  in  the  large  square  there  are 
sixteen  small  squares,  the  sides  of  which  are  each  one  foot. 
Therefore,  the  square  whose  side*  is  four  feet,  contains  six- 
teen square  feet. 

The  number  of  small  squares  that  is  contained  in  any  large 
square  is  always  equal  to  the  product  of  two  of  the  sides  of 
the  large  square.  As  in  the  figure,  4x4=^16  square  feet. 
The  number  of  square  inches  contained  in  a  square  foot  is 
equal  to  12X12=144. 

326.  A  triangle  is  a  figure  bounded  by  three  straight  lines. 
Thus,  BAG  is  a  triangle. 

Quest. — 324.  What  is  mensuration  ?  325.  What  is  a  surface  ?  What 
fe  a  square?  What  is  the  number  of  small  squares  contained  in  a 
large  square  equal  to  ?     326.  What   is   a   triangle  ? 


S40 


MENSURATION. 


The  three  lines  BA,  AC,  BC,  are  call-      , g 

ed  sides:  and  the  three  corners,  B,  A, 
and  C,  are  called  angles.  The  side  AB 
is  called  the  base. 

"When  a  line  like  CD  is  drawn  making     A  D  "B 

the  angle  CDA  equal  to  the  angle  CDB,  then  CD  is  said  to 
be  perpendicular  to  AB,  and  CD  is  called  the  altitude  of  the 
triangle.  Each  triangle  CAD  or  CDB  is  called  a  right-angled 
triangle.  The  side  BC,  or  the  side  AC,  opposite  the  right 
angle,  is  called  the  hypothenuse. 

The  area  or  contents  of  a  triangle  is  equal  to  half  the  jpro^ 
duct  of  its  base  by  its  altitude  (Bk.  IV.  Prop.  VI).* 

EXAMPLES. 

1.  The  base,  AB,  of  a  triangle  is  50  yards,  and  the  per- 
pendicular, CD,   30  yards:  what  is  the  area? 

OPERATION. 


We  first  multiply  the  base 
by  the  altitude,  and  the  pro- 
duct is  square  yards,  which 
we  divide  by   2  for  the  area. 


.    50 

30 

2)1500 
Ans.     75 0   square  yards. 


2.  In  a  triangular  field  the  base  is  60  chains,  and  the  per- 
pendicular 12  chains  :  how  jnuch  does  it  contain  ? 

3.  There  is  a  triangular  field,  of  which  the  base  is  45  rods 
and  the  perpendicular  38  rods  :  what  are  its  contents  ? 

4.  What  are  the  contents  of  a  triangle  whose  base  is  75  chains 
and  perpendicular  36  chains  ? 

327.  A  rectangle  is  a  four-sided  figure  like 
a  square,  in  which  the  sides  are  perpendicular 
to  each  other,  but  the  adjacent  sides  are  not 
equal. 

*  All  the  references  are  to  Da  vies'  Legendre. 

Quest. — 326.  Whet  is  the  base  of  a  triangle?  What  the  altitude? 
What  is  a  right-angled  triangle  ?  Which  side  is  the  hypothenuse  ? 
What  is  the  area  of  a  triangle  equal  to  ?     327.  What  is  a  rectangle  ? 


MENSURATION.  841 

328.  A  parallelogram  is  a  four-sided 
figure  which  has  its  opposite  sides  equal 
and  parallel,  but  its  angles  not  right- 
angles.  The  line  DE,  perpendicular  to 
the  base,  is  called  the  altitude.  ^ 

329.  To  find  the  area  of  a  square,  rectangle,  or  parallelo* 
gram, 

Multiply  the  base  hy  the  'perpendicular  height^  and  the  pro- 
duct will  he  the  area  (Bk.  IV.   Prop.  V). 

EXAMPLES. 

1.  What  is  the  area  of  a  square  field  of  which  the  sides  are 
each  66.16  chains  ? 

2.  What  is  the  area  of  a  square  piece  of  land  of  which  the 
sides  are  54  chains? 

3.  What  is  the  area  of  a  square  piece  of  land  of  which  the 
sides  are  75  rods  each  ? 

4.  What  are  the  contents  of  a  rectangular  field,  the  length 
of  which  is  80  rods  and  the  breadth  40  rods  ? 

5.  What  are  the  contents  of  a  field  80  rods  square  ? 

6.  What  are  the  contents  of  a  rectangular  field  30  chains 
long  and  5  chains  broad  ? 

Y.  What  are  the  contents  of  a  field  54  chains  long  and 
1 8  rods  broad  ? 

8.  The  base  of  a  parallelogram  is  542  yards,  and  the  per- 
pendicular height  720  feet :  what  is  the  area  ? 

330.  A  trapezoid  is  a  four-sided  figure  DEO 
ABCD,  having  two  of  its  sides,  AB,  DC, 
parallel.     The  perpendicular  EF  is  called 


the  altitude.  A         F  B 


Quest. — 328.  What  is  a  parallelogram?  829.  How  do  you  find  the 
area  of  a  square,  rectangle,  or  parallelogram  ?  880.  What  is  a  trape- 
zoid ? 


342  MENSU^Affo'*?f 

331.    To  find  the  area  of  a  trapezoid, 

Multiply  the  sum  of  the  two  parallel  sides  hy  the  altitudCy 
and  divide  the  product  by  2,  and  the  quotient  will  be  the  area 
(Bk.  IV.  Prop.  VII). 

EXAMPLES. 

1.  Required  the  area  or  contents  of  the  trapezoid  ABCD, 
having  given  AB= 643.02  feet,  DC=428.48  feet,  and  EF 
=  342.32  feet. 

We  first  find  the  sum  of 
the  sides,  and  then  mul- 
tiply it  by  the  perpendi- 
cular height,  after  which, , 
we  divide  the  product  by 
2,  for  the  area. 


OPERATION. 

643.02  +  428.48  =  1071.60,  = 
sum  of  parallel  sides.  Then, 
1071.50X  342.32  =  366795.88; 

and,   3667^95.88-^i3339>794^ 

the  area. 


2.  What  is  the  area  of  a  trapezoid,  the  parallel  sides  of 
which  are  24.82  and  16.44  chains,  and  the  perpendicular  dis- 
tance between  them   10.30  chains? 

3.  Required  the   area  of  a   trapezoid  whose   parallel  sides 
'  are  51  feet,  and  37  feet  6  inches,  and  the  perpendicular  dis- 
tance between  them  20  feet  10  inches. 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  41  and  24.5,  and  the  perpendicular  distance  between 
them  21.5  yards. 

5.  Wliat  is  the  area  of  a  trapezoid  whose  parallel  sides  are 
15  chains,  and  24.5  chains,  and  the  perpendicular  height 
80.8  chains? 

6.  What  are  the  contents  when  the  parallel  sides  are  40  and 
64  chains,  and  the  perpendicular  distance  between  them  52 
chains  ? 

Quest.-— 331.  How  do  you  find  the  area  of  a  trapezoid? 


MENSURATION. 


343 


332.  A  circle  is  a  portion  of  a  plane 
bounded  by  a  curved  line,  every  part  of 
wliicli  is  equally  distant  from  a  certain 
point  within,  called  the  centre. 

The  curved   line   AEBD  is   called   the 
circumference;  the  point  C  the  centre;  the 
line  AB  passing  through  the  centre,  a  diameter ;  and  CB  the 
radius. 

The  circumference  AEBD  is  3.1416  times  as  great  as  the 
diameter  AB.  Hence,  if  the  diameter  is  1,  the  circumference 
will  be  3.1416.  Therefore,  if  the  diameter  is  known,  the  cir- 
cumference is  found  by  multiplying  3.1416  by  the  diameter 
(Bk.  V.  Prop.  XIV). 


EXAMPLES. 

1.   The  diameter  of  a  circle  is  8  :   what  is  the  circumference  ? 

The  circumference  is  found  by  3  1416* 

simply  multiplying  3.1416  by  the  8 

diameter.  j^ris,  25.1328 


what  is  the  circumfer- 
what  is  the  circumfer- 


2.   The  diameter  of  a  circle  is  186 
ence  ? 

8.   The  diameter  of  a  circle  is  40  ; 
ence  ? 

4.  What  is  the  circumference  of  a  circle  whose  diameter 
is  57? 

333.  Since  the  circumference  of  a  circle  is  3.1416  times 
as  great  as  the  diameter,  it  follows,  that  if  the  circumference  is 
known,  we  may  find  the  diameter  by  dividing  it  by  3.1416. 


Quest. — 332.  What  is  a  circle?  What  is  the  centre?  What  is  the 
circumference  ?  What  is  the  diameter  ?  What  the  radius  ?  How  many 
times  greater  is  the  circumference  than  the  diameter  ?  How  do  you 
find  the  circumference  when  the  diameter  is  known  ?  333.  How  do 
you  find   the   diameter  when  the  circumferftnce  is  known  ? 


344 


MENSURATION. 


OPERATION. 

3.1416)157.080(50 
15*7.080 


EXAMPLES. 

1.  What  is  the  diameter  of  a  circle   whose  circumference 
is  157.08? 

We  divide  the  circumference 
by  3.1416,  the  quotient  50  is  the 
diameter. 

2.  What  is  the  diameter  of  a  circle  whose  circumference 
is  23304.3888? 

3.  What  is  the  diameter  of  a  circle  whose  circumference 
is  13700?  Ans,  . 

334.  To  find  the  area  or  contents  of  a  circle, 

Multiply  the  square  of  the  diameter  hy  the  decimal  .7854 
(Bk.  V.  Prop.  XII.  Cor.  2). 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  12  ? 

We  first  square  the  diam- 
eter, giving  144,  which  we 
then  multiply  by  the  decimal 
.7854 :  the  product  is  the 
area  of  the  circle. 

2.  What  is  the  area  of  a  circle  whose  diameter  is  5  ? 

3.  What  is  the  area  of  a  circle  whose  diameter  is  14  ? 

4.  How  many  square  yards  in  a  circle  whose  diameter  is 
3^  feet? 

335.  x\  sphere  is  a  solid  termina- 
ted by  a  curved  surface,  all  the  points 
of  which  are  equally  distant  from  a 
certain  point  within,  called  the  centre. 
The  hne  AD,  passing  through  its  cen- 
tre C,  is  called  the  diameter  of  the 
sphere,  and  AC  its  radius. 

Quest. — 834.  How  do  you  find  the   area  of  a  circle  ?      385.  What 
is  a  sphere  ?     "What  is  a  diameter  ?     What  is  a  radius  ? 


OPERATION. 

T?=144 
144  X. 7854=113.0976 
Ans.    113.0976 


MENSURATION. 


345 


336.    To  find  the  surface  of  a  sphere, 
Multiply  the  square  of  the  diameter  hy  3.1416  (Bk.  VIII, 
Prop.  X.    Cor.) 

EXAMPLES. 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  6  ? 

We  simply  multiply  the  deci- 
mal 3.1416  by  the  square  of 
the  diameter :  the  product  is  the 
surface. 

2.  What  is  the  surface  of  a  sphere  whose  diameter  is  14  ? 

3.  Required  the  number  of  square  inches  in  the  surface 
of  a  sphere  whose  diameter  is   3   feet  or  36   inches. 

4.  Required  the  area  of  the  surface  of  the  earth,  its  mean 
diameter  being  79 18.7  miles. 


OPERATION. 

3.1416 

6'= 36 

Arts.     113.976 


8  feet==rl  yard. 


MENSURATION  OF  SOLIDS. 
337.  A  cube  is  a  body,  or  sohd, 
having  six  equal  faces,  ^ijjiich  are 
squares.  If  the  sides  of  the  cube 
be  each  one  foot  long,  the  solid  is 
called  a  cubic  or  solid  foot.  But 
when  the  sides  of  the  cube  are  one 
yard,  as  in  the  figure,  the  cube  is 
called  a  cubic  or  solid  yard.  The  base  of  the  cube,  which  is 
the  face  on  which  it  stands,  contains  3x3  =  9  square  feet. 
Therefore  9  cubes,  of  one  foot  each,  can  be  placed  on  the  base. 
If  the  sohd  were  one  foot  high  it  would  contain  9  cubic  feet; 
if  it  were  2  feet  high  it  would  contain  two  tiers  of  cubes,  or 
18  cubic  feet;  and  if  it  were   3  feet  high,  it  would  contain 

Quest. — 836.  How  do  you  find  the  surface  of  a  sphere  ?  837.  What 
is  a  cube  ?  What  is  a  cubic  or  solid  foot  ?  What  is  a  cubic  yard  ?  How 
many  cubic  feet  in  a  cubic  vard  ? 

IS* 


346 


MENSURATION. 


three  tiers,  or  27  cubic  feet.  Hence,  the  contents  of  a  solid 
are  equal  to  the  product  of  its  length,   breadth,  and  height, 

338.  To  find  the  solidity  of  a  sphere, 

Multiply  the  surface  by  the  diameter  and  divide  the  pro- 
duct by  6,  the  quotient  will  be  the  solidity  (Bk.  VIII.  Prop. 
XIV.  Sch.  3). 

EXAMPLES. 

1.  What  is  the  sohdity  of  a  sphere  whose  diameter  is  12  ? 

OPERATION. 


We  first  find  the  surface  by 
multipljdng  the  square  of  the 
diameter  by  3.1416.  We  then 
multiply  the  surface  by  the  dia- 
meter, and  divide  the  product 
by  6. 


12=144 
multiply  by  3.1416 
surface  =452.3904 

diameter  12 

6)5428.6848 
sohdity         =904.'7808 


2.  What  is  the  solidity  of  a  sphere  whose  diameter  is  8  ? 

3.  What  is  the  solidity  of  a  sphere  whose  diameter  is  16 
inches  ? 

4.  What  is  the  sohdity  of  the  earth,  its  mean  diameter  be- 
ing 7918.7  miles? 

6.  What  is  the  solidity  of  a  sphere  whose  diameter  is  12  feet? 

339.  A  prism  is  a  soHd  whose  ends 
are  equal  plane  figures  and  whose 
faces  are  parallelograms. 

The  sum  of  the  sides  which  bound 
the  base  is  called  the  perimeter  of  the 
base,  and  the  sum  of  the  parallelo- 
grams which  bound  the  solid  is  called 
the  convex  surface. 

340.  To  find  the  convex  surface  of 
a  right  prism. 

Quest. — What  are  the  contents  of  a  solid  equal  to?  838.  How  do 
you  find  the  solidity  of  a  sphere  ?  339.  What  is  a  prism  ?  Wliat  is 
th*?  perimeter  of  the  ba'^e  ?      What   is  the  convex  surface  ? 


MENSURATION.  347 

Multiply  the  perimeter  of  the  base  hy  the  perpendicular 
height^  and  the  product  will  he  the  convex  surface  (Bk.  VII. 
Prop.  I). 

EXAMPLES. 

1.  What  is  the  convex  surface  of  a  prism  whose  base  is 
oounded  by  five  equal  sides,  each  of  which  is  35  feet,  the  alti- 
tude being  52  feet? 

2.  What  is  the  convex  surface  when  there  are  eight  equal 
sides,  each  15  feet  in  length,  and  the  altitude  is  12  feet  ? 

341..    To  find  the  solid  contents  of  a  prism. 

Multiply  the  area  of  the  base  by  the  altitude,  and  the 
prodvAH  will  be  the  contents   (Bk.  VII.  Prop.  XIV). 

EXAMPLES. 

1.  What  are  the  contents  of  a  square  prism,  each  side 
of  the  square  which  forms  the  base  being  16,  and  the  alti- 
tude of  the  prism  30  feet  ? 

We  first  find   the  area  of  the 
square  which  forms  the  base,  and 
.  then  multiply  by  the  altitude. 


16  =256 
30 


Ans.    7680 

'2.  What  are  the  solid  contents  of  a  cube,  each  side  of  which, 
is  48  inches  ? 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which 
the  length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and  height 
or  thickness   5  feet  ? 

4.  How  many  gallons  of  water  will  a  cistern  contain,  whose 
dimensions  are  the  same  as  in  the  last  example  ? 

5.  Required  the  soUdity  of  a  triangular  prism,  whose  height 
is  20  feet,  and  area  of  the  base  691. 

Quest. — 340.  How  do  you  find  the  convex  surface  of  a  prism? 
841.  How  do  you  find  the  sohd  contents  of  a  prism  ? 


848 


MENSURATION. 


342.  A  cylinder  is  a  round  body  with 
circular  ends.  The  line  EF  is  called  the 
axis  or  altitude,  and  the  circular  surface 
the  convex  surface  of  the  cyhnder. 


343.    To  find  the  convex  surface  of  a  cylinder, 
Multiply   the   circumference   of   the   base    by    the   altitude, 

and    the   product    will    be    the    convex    surface    (Bk.  VIII. 

Prop.   I). 

EXAMPLES. 

1.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  20  and  the  altitude  40  ? 


We  first  multiply  3.1416  by 
the  diameter,  which  gives  the  cir- 
cumference of  the  base.  Then 
multiplying  by  the  altitude,  we 
obtain  the  convex  surface. 


OPERATION. 

3.1416 
20 


62.8320 

40 

Ans.    2513.2800 


2.  AVhat  is  the  convex  surface  of  a  cylinder  whose  altitude 
is  28  feet  and  the  circumference  of  its  base  8  feet  4  inches  ? 

3.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  15  inches  and  altitude  5  feet  ? 

4.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  40  and  altitude  50  feet  ? 

344.  To  find  the  solidity  of  a  cylinder. 
Multiply  the  area  of  the  base  by  the  altitude :  the  product 
will  be   the  solid  contents  (Bk.  VIII.  Prop.  11). 

Quest. — 342.  "What  is  a  cylinder  ?  What  is  the  axis  or  altitude  ? 
"What  is  the  convex  surface  ?  343.  How  do  you  find  the  convex  surface  ? 
844.  How  do  you  find  the  solidity  of  a  cvlinder  ? 


MENSURATION. 


349 


EXAMPLES. 

1.  Required  the  solidity  of  a  cylinder  of  wliicli  the  altitude 
is  11  feet,   and  the  diameter  of  the  base  16  feet. 

OPERATION. 

"We  first  find  the  area  of  the 
base,  and  then  multiply  by  the 
altitude :  the  product  is  the  soli-  area  base 

dity. 


16 


256 

.7854 

201.0624 

11 

2111.6864 


2.  What  is  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  40  and  the  altitude  29  ? 

3.  WTiat  is  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  24  and  the  altitude  30  ? 

4.  What  is  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  32  and  altitude  12? 

5.  What  is  the  solidity  of  a  cylinder,  the  diameter  of  whose 
base  is  25  and  altitude  15  ? 


345.  A  pyramid  is  a  solid  formed 
by  several  triangular  planes  united  at 
the  same  point  S,  and  terminating  in 
the  different  sides  of  a  plane  figure,  as 
ABODE.  The  altitude  of  the  pyramid 
is  the  line  SO,  drawn  perpendicular  to 
the  base.  A- 


346.  To  find  the  solidity  of  a  pyramid. 
Multiply  the  area  of  the  base  by  the  altitude^  and  divide 
the  product  by  3  (Bk.  VII.  Prop.  XVII). 


Quest. — 345.  What  is  a  pyramid  ?    What  is  the  altitude  of  a  pyramid  ? 
846.  How  do  you  find  the  solidity  of  a  pyramid  ? 


350 


MENSURATION. 


EXAMPLES. 


1.  Required  the  solidity  of  a  pyramid,  of  wMcli  tlie  area 


of  the  base  is  86  and  the  altitude  24. 

We  simply  multiply  the  area  of  the 
base  86,  by  the  altitude  24,  and  then 
divide  the  product  by  3. 


OPERATION. 

86 
24 


344 
172 

3)2064 
Arts,     688 


2.  What  is  the  soHdity  of  a  pyramid,  the  area  of  whose 
base  is  365  and  the  altitude  36  ? 

3.  What  is  the  solidity  of  a  pyramid,  the  area  of  whose 
base  is  207  and  altitude  36  ? 

4.  What  is  the  sohdity  of  a  pyramid,  the  area  of  whose 
base  is  562  and   altitude  30  ? 

5.  What  are  the  solid  contents  of  a  pyramid,  the  area  of 
whose  base  is  540  and  altitude  32  ? 

6.  A  pyramid  has  a  rectangular  base,  the  sides  of  which 
are  50  and  24 ;  the  altitude  of  the  pyramid  is  36 :  what 
are  its  solid  contents  ? 

7.  A  pyramid  with  a  square  base,  of  which  each  side 
is  15,  has  an  altitude  of  24:  what  are  its  solid  contents? 

C 


347.  A  cone  is  a  round  body  with 
a  circular  base,  and  tapering  to  a  point 
called  the  vertex.  The  point  C  is  the 
vertex,  and  the  line  CB  is  called  the 
axis  or  altitude. 


Quest. — 347.  What  is  a  cone?     What  is  the  vertex?     What  is  the 
axis  ?     348.  How  do  you  find  the  s<^dity  of  a  cone  ? 


MENSURATION. 


351 


348.   To  find  the   solidity  of  a  cone, 

Multiply  the  area  of  the  base  by  the  altitude,  and  divide 
the  product  by  3 ;  or,  multiply  the  area  of  the  base  by  one- 
third  of  the  altitude,     (Bk.  VIII.   Prop.  V.) 

EXAMPLES. 

1.   Required  the  solidity  of  a  cone,  the  diameter  of  whose 
is  6   and  the  altitude   11. 

OPERATION, 

36x.V854  =  28.2744 
11 


3)311.0184 
Ans.     103.6728 


We  first  square  the  diameter, 
and  multiply  it  by  .7854,  which 
gives  the  area  of  the  base.  We 
next  multiply  by  the  altitude,  and 
then  divide  the  product  by  3. 

2.  What  is  the  solidity  of  a  cone,  the  diameter  of  whose 
base  is  36  and  the  altitude  27? 

3.  What  are  the   solid  contents  of  a  cone,    the    diameter 
of  whose  base  is  35  and  the  altitude  27  ? 

4.  What  is  the  solidity  of  a  cone,  whose  altitude  is  27  feet 
and  the  diameter  of  the  base  20  feet? 

RIGHT  ANGLED   TRIAI^GLE. 

349.  The  properties  of  the  right  angled  are  so  important 
as  to  be  worthy  of  particular  notice. 

In  every  right  angled  tri- 
angle, the  square  described  on 
the  hypothenuse,  is  equal  to 
the  sum  of  the  squares  de- 
scribed on  the  other  two  sides. 

Thus,  if  ABC  be  a  right  an- 
gled triangle,  right  angled  at 
C,  then  will  the  square  D  de- 
scribed on  AB  be  equal  to  the 
sum  of  the  squares  E  and  F, 
described  on  the  sides  CB  and 
AC.     This  is  called  the  carpenter's  theorem. 


~ 

D 

352  MENSURATION.     # 

Hence,  to  find  the  liypothenuse  when  the  base  and  per- 
pendicular are  known, 

1st.  Square  each  side  separately,  2d.  Add  the  squares 
together.  3d.  Extract  the  square  root  of  the  sum^  and  the 
result  will  he   the  hypothenuse  of  the  triangle, 

EXAMPLES. 

1.  The  wall  of  a  building,  on  the  brink  of  a  river  is  120 
feet  high,  and  the  breadth  of  the  river  VO  yards :  what  is  the 
length  of  a  line  which  would  reach  from  the  top  of  the  wall 
to  the  opposite  edge  of  the  river  ? 

2.  The  side  roofs  of  a  house  of  which  the  eaves  are  of  the 
same  height,  form  a  right  angle  with  each  other  at  the  top. 
Now,  the  length  of  the  rafters  on  one  side  is  ItT  feet,  and  on 
the  other  14  feet :  what  is  the  breadth  of  the  house  ? 

3.  What  would  be  the  width  of  the  house,  in  the  last  ex- 
ample, if  the  rafters  on  each  side  were  10  feet  ? 

350.  When  the  hypothenuse  and  one  side  of  a  right  an- 
gled triangle  are  known,  to  find  the  other  side. 

Square  the  hypothenuse  and  also  the  other  given  side^  and 
take  their  difference :  extract  the  square  root  of  their  differ 
ence,  and  the  result  will  he   the    required  side, 

1.  The  height  of  a  precipice  on  the  brink  of  a  river  is  10*3 
feet,  and  a  line  of  320  feet  in  length  will  just  reach  from  the 
top  of  it  to  the  opposite  bank :  required  the  breadth  of  the 
river. 

2.  The  hypothenuse  of  a  triangle  is  53  yards,  and  the  per- 
pendicular 45  yards:  what  is  the  base? 

3.  A  ladder  60  feet  in  length,  will  reach  to  a  window  40 
feet  from  the  ground  on  one  side  of  the  street,  and  by  turn- 
ing it  over  to  the  other  side,  it  will  reach  a  window  50  feet 
from  the  ground:   required  the  breadth  of  the  street. 

Quest. — 349.  What  is  the  property  of  a  right  angled  triangle  ?  "When 
can  you  find  the  hypothenuse?    How  ?     350.  How  do  you  find  a  side  ? 


OF    THE    MECHANICAL    POWERS.  353 


OF  THE  MECHANICAL  POWERS  * 

351.  There  are  six  simple  machines,  which  are  called  Me- 
chanical powers.  They  are,  the  Lever,  the  Pulley,  the  Wheel 
and  Axle,  the  Inclined  Plane,  the  Wedge,  and  the  Screw. 

352.  To  understand  the  power  of  a  machine,  four  things 
must  be  considered. 

1st.  The  power  or  force  which  acts.  This  consists  in  the 
efforts  of  men  or  horses,  of  weights,  springs,  steam,  <fec. 

2d.  The  resistance  which  is  to  be  overcome  by  the  power. 
This  generally  is  a  weight  to  be  moved. 

3d.  We  are  to  consider  the  centre  of  motion,  or  fulcrum^ 
which  means  a  prop.  The  prop  or  fulcrum  is  the  point  about 
which  all  the  parts  of  the  machine  move. 

4th.  We  are  to  consider  the  respective  velocities  of  the 
power  and  resistance. 

353.  A  machine  -is  said  to  be  in  equilibrium  when  the 
resistance  exactly  balances  the  power,  in  which  case  all  the 
parts  of  the  machine  are  at  rest. 

We  shall  first  examine  the  lever. 

354.  The  Lever,  is  a  straight  bar  of  wood  or  metal,  which 
moves  around  a  fixed  point,  called  the  fulcrum.  There  are 
three  kinds  of  levers. 


1st.  When   the  fulcrum    is    be- 
tween the  weight  and  the  power. 


i 


*  This  article  is  taken  from  a  Practical  Work  for  mechanics,  entitled 
"  Mensuration  and  Drawing." 

Quest. — 351.  How  many  simple  machines  are  there  ?  What  are  they 
called  ?  352.  What  things  must  be  considered  in  order  to  understand  the 
power  of  a  machine  ?  853.  When  is  a  machine  said  to  be  in  equili- 
brium ?  854.  What  is  a  lever  ?  How  many  kinds  of  levers  are  there  ? 
Describe  the  first  kind. 


354 


OP    THE    MECHANICAL    POWERS. 


2d.  When  the  weight  is 
between  the  power  and-  the 
fulcrum. 


3d.  When  the  power  is 
between  the  fulcrum  and  the 
weight.  ^ 

The  parts  of  the  lever 
from  the  fulcrum  to  the 
weight  and  power,  are  call- 
ed the  arms  of  the^  lever. 

355.  An  equilibrium  is  produced  in  all  the  levers,  when 
the  weight  multiplied  by  its  distance  from  the  fulcrum  is 
equal  to  the  product  of  the  power  multiplied  by  its  distance 
from  the  fulcrum.     That  is. 

The  weight  is  to  the  power ,  as  the  distance  from  the  power 
to  the  fulcrum,  is  to  the  distance  from  the  weight  to  the 
fulcrum. 

♦      EXAMPLlES. 

1.  In  a  lever  of  the  first  kind,  the  fulcrum  is  placed  at 
the  middle  point :  what  power  will  be  necessary  to  balance  a 
weight  of  40  pounds  ? 

2.  In  a  lever  of  the  second  kind,  the  weight  is  placed  at 
the  middle  point :  what  power  will  be  necessary  to  sustain 
a  weight  of  50  lbs.  ? 

3.  In  a  lever  of  the   third   kind,  the  power  is  placed  at 


Quest. — Where  is  the  weight  placed  in  the  second  kind?  Where 
is  the  power  placed  in  the  third  kind  ?  855.  When  is  an  equilibrium 
produced  in  all  the  levers  ?  What  is  then  the  proportion  between  the 
weight  and  power? 


OF    THE    MECHANICAL    POWERS.  355 

the  middle  point:   what  power  will  be  necessary  to  sustain 
a  weight  of  25  lbs.? 

4.  A  lever  of  the  first  kind  is  8  feet  long,  and  a  weight 
of  60  lbs.  is  at  a  distance  of  2  feet  from  the  fulcrum :  what 
power  will  be  necessary  to  balance  it  ? 

5.  In  a  lever  of  the  first  kind,  that  is  6  feet  long,  a  weight 
of  200  lbs.  is  placed  at  1  foot  from  the  fulcrum  :  what  power 
will  balance  it? 

6.  In  a  lever  of  the  first  kind,  like  the  common  steelyard, 
the  distance  from  the  weight  to  the  fulcrum  is  one  inch: 
at  what  distance  from  the  fulcrum  must  the  poise  of  1  lb. 
be  placed,  to  balance  a  weight  of  1  lb.  ?  A  weight  of  1^  lbs.? 
Of  2  lbs.  ?     Of  4  lbs.  ? 

v.  In  a  lever  of  the  third  kind,  the  distance  from  the 
fulcrum  to  the  power  is  5  feet,  and  from  the  fulcrum  to  the 
.weight  8  feet:  what  power  is  necessary  to  sustain  a  weight 
of. 40  lbs.? 

8.  In  a  lever  of  the  third  kind,  the  distance  from  the  ful- 
crum to  the  weight  is  12  feet,  and  to  the  power  8  feet: 
what  power  will  be  necessary  to  sustain  a  weight  of  100  lbs.? 

356.  Remarks. — In  determining  the  equilibrium  of  the 
lever,  we  have  not  considered  its  weight.  In  levers  of  the 
first  kind,  the  weight  of  the  lever  generally  adds  to  the 
power,  but  in  the  second  and  third  kinds,  the  weight  goes  to 
diminish  the  effect  of  the  power. 

In  the  previous  examples,  we  have  stated  the  circumstances 
under  which  the  power  will  exactly  sustain  the  weight.  In 
order  that  the  power  may  overcome  the  resistance,  it  must 
of  course  be  somewhat  increased.  The  lever  is  a  very  im- 
portant mechanical  power,  being  much  used,  and  entering 
indeed  into  all  the  other  machines. 

Quest. — 356.  Has  the  weight  been  considered  in  determining  the  equi- 
librium of  the  levers?  In  a  lever  of  the  first  kind,  will  the  weight 
increase  or  diminish  the  power  ?     How  will  it  be  in  the  two  other  kinds  ? 


356 


OF    THE    MECHANICAL    POWERS. 


OF    THE    PULLEY. 

357.  The  pulley  is  a  wheel,  having  a 
groove  cut  in  its  circumference,  for  the 
purpose  of  receiving  a  cord  which  passes 
over  it.  When  motion  is  imparted  to  the 
cord,  the  pulley  turns  around  its  axis, 
which  is  generally  supported  by  being  at- 
tached to  a  beam  above. 

358.  Pulleys  are  divided  into  two  kinds,  fixed  pulleys  and 
moveable  pulleys.  When  the  pulley  is  fixed,  it  does  not 
increase  the  power  which  is  applied  to  raise  the  weight,  but 
merely  changes  the  direction  in  which  it  acts. 


359.  A  moveable  pulley  gives  a  mechan- 
ical advantage.  Thus,  in  the  moveable 
pulley,  the  hand  which  sustains  the  cask 
does  not  actually  support  but  one-half  the 
weight  of  it;  the  other  half  is  supported 
by  the  hook  to  which  the  other  end  of 
the  cord  is  attached. 


360.  If  we  have  several  moveable  pulleys,  the  advantage 
gained  is  still  greater,  and  a  very  heavy  weight  may  be 
raised  by  a  small  power.  A  longer  time,  however,  will  be 
required,  than  with  the  single  pulley.  It  is  indeed  a  general 
principle   in    machines,    that   what   is    gained    in    power,    is 


Quest. — 357.  What  is  a  pulley  ?  858.  How  many  kinds  of  pulleys 
are  there  ?  Does  a  fixed  pulley  give  any  increase  of  power  ?  859.  Does 
a  moveable  pulley  give  any  mechanical  advantage  ?  In  a  single  move- 
able pulley,  how  much  less  is  the  power  than  the  weight  ?  860.  Will 
an  advantage  be  gained  by  several  moveable  pulleys? 


OP    THE    MECHANICAL    POWERS. 


357 


A 


^ 


lost  in  time;  and  this  is  true  for  all  ma- 
chines. There  is  also  an  actual  loss  of  power, 
viz.  the  resistance  of  the  machine  to  mo- 
tion, arising  from  the  rubbing  of  the  parts 
against  each  other,  which  is  called  the  friction 
of  the  machine.  This  varies  in  the  different 
machines,  but  must  always  be  allowed  for,  in 
calculating  the  power  necessary  to  do  a  given 
work.  It  would  be  wrong,  however,  to  sup- 
pose that  the  loss  was  equivalent  to  the  gain, 
and  that  no  advantage  i*  derived  from  the  me- 
chanical powers.  We  are  unable  to  augment 
our  strength,  but,  by  the  aid  of  science  we  so 
divide  the  resistance,  that  by  a  continued  exer-        I  1 

tion  of  power,  we  accomplish  that  which  it 
would  be  impossible  to  effect  by  a  single  effort. 

If  in  attaining  this  result,  we  sacrifice  time,  we  cannot 
but  see  that  it  is  most  advantageously  exchanged  for  power. 

361.  It  is  plain,  that  in  the  moveable  pulley,  all  the  parts, 
of  the  cord  will  be  equally  stretched,  and  hence,  each  cord 
running  from  pulley  to  pulley,  will  bear  an  equal  part  of  the 
weight;  consequently  the  power  will  always  he  equal  to  the 
weighty  divided  hy  the  number  of  cords  which  reach  from 
pulley  to  pulley, 

EXAMPLES. 

1.  In  a  single  immoveable  pulley,  what  power  will  support 
a  weight   of  60  lbs.? 

2.  In  a  single  moveable  pulley,  what  power  will  support 
a  weight  of  80  lbs.? 

3.  In  two  moveable  pulleys  with  5  cords,  (see  last  fig.,) 
what  power  will  support  a  weight  of  100  lbs.  ? 


Quest. — State  the  general  principle  in  machines.  What  does  the 
actual  loss  of  power  arise  from  ?  What  is  this  rubbing  called  ?  Does 
this  vary  in  different  machines?  361.  In  the  moveable  pulley,  what 
proportion   exists  between  the  cord  and  the  weight  ? 


358  OF    THE    MECHANICAL    POWERS. 

WHEEL     AND     AXLE. 

362.  This  machine  is  com-  ,.-., 
posed  of  a  wheel   or  crank 

— firmly  attached  to  a  cyl-  _.«,.^^^ 

indrical  axle.     The    axle    is  r^^^^^^^^^^k.     '• 

supported  at  its  ends  by  two  r^^^^^^^^^v  m   / 

pivots,  which  are  of  less  dia-  i^k\  Im  ^^^^^ 

meter  than  the  axle  around         /^^  ^'-'-r:^J^^J  I  W^ 
which    the    rope   is    coiled,    <JI  ^^^^^^'^^^^^^^^ 
and  which  turn  freely  about      /  ^  Jm^^a  ^^^^^ 

the   points   of  support.     In    ^^^'^^^^M,^^-<^^^^^ 
order  to  balance  the  weight,  x:is#^^^^^^^^^ 

we  must  have 

The  power  to  the  weighty  as  the  radius  of  the  axle^  to  the 
length  of  the  cranky  or  radius  of  the  wheel, 

EXAMPLES. 

1.  What  must  be  the  length  of  a  crank  or  radius  of  ^ 
wheel,  in  order  that  a  power  of  40  lbs.  may  balance  a  weight 
of  600  lbs.  suspended  from  an  axle  of  6  inches  radius  ? 

2.  What  must  be  the  diameter  of  an  axle  that  a  power  of 
100  lbs.  applied  at  the  circumference  of  a  wheel  of  6  feet 
diameter  may  balance  400  lbs.! 

INCLINED     PLANE. 

363.  The  inclined  plane  is  nothing  more  than  a  slope  or 
declivity,  which  is  used  for  the  purpose  of  raising  weights. 
It  is  not  difficult  to  see  that  a  weight  can  be  forced  up  an 
inclined  plane,  more  easily  than  it  can  be  raised  in  a  vertical 
line.  But  in  this,  as  in  the  other  machines,  the  advantage 
is  obtained  by  a   partial  loss  of  power. 

Quest. — 362.  Of  what  is  the  machine  called  the  wheel  and  axle,  com- 
posed ?  How  is  the  axle  supported  ?  Give  the  proportion  between  the 
power  and  the  weight.     363.  What  is  an  inclined  plane  ? 


OF    THE    MECHANICAL    POWERS. 


359 


Thus,  if  a  weight  W, 
be  supported  on  the  in- 
clined plane  ABC,  by  a 
cord  passing  over  a  pul- 
ley at  F,  and  the  cord 
from  the  pulley  to  the  weight  be  parallel  to  the  length  of 
the  plane  AB,  the  power  P,  will  balance  the  weight  W,  when 
P  :  W  :  :  height  BG  :  length  AB. 

It  is  evident  that   the  power    ought   to  be  less  than  the 
weight,  since  a  part  of  the  weight  is  supported  by  the  plane. 

EXAMPLES. 

1.  The  length  of  a  plane  is  30  feet,  and  its  height  6  feet: 
what  power  will  be  necessary  to  balance  a  weight  of  200  lbs.? 

2.  The  height  of  a  plane  is  10  feet,  and  the  length  20  feet: 
what  weight  will  a  power  of  50  lbs.  support? 

3.  The  height  of  a  plane  is  15   feet,  and  length  45  feet: 
what  power,  will  sustain  a  weight  of  180  lbs.? 


THE    WEDGE. 

364.  The  wedge  is  composed  of  two 
inclined  planes,  united  together  along 
their  bases,  and  forming  a  solid  ACB. 
It  is  used  to  cleave  masses  of  wood  or 
Btone.  The  resistance  which  it  over- 
comes is  the  attraction  of  cohesion  of 
the  body  which  it  is  employed  to  separate.  The  wedge  acts 
principally  by  being  struck  with  a  hammer,  or  mallet,  on  its 
head,  and  very  little  effect  can  be  produced  with  it,  by  mere 
pressure. 

All   cutting  instruments    are   constructed   on  the    principle 

Quest. — What  proportion  exists  between  the  power  and  weight  when 
they  are  in  equilibrium  ?  864.  What  is  the  wedge  ?  What  is  it  used 
*  )r  ?     What  resistance  is  it  used  to  overcome  ? 


360 


OF    THE    MECHANICAL    POWERS. 


of  the  inclined  plane  or  wedge.  Sucli  as  have  but  one  slop- 
ing edge,  like  the  chisel,  may  be  referred  to  the  inclined  plane, 
and  such  as  have  two,  like  the  axe  and  the  knife,  to  the 
wedge. 

THE     SCREW. 

365.  The  screw  is  composed 
of  two  parts — the  screw  S,  and 
the  nut  ]Sr. 

The  screw  S,  is  a  cylinder 
with  a  spiral  projection  wind- 
ing around  it,  called  the 
thread.  The  nut  N  is  per- 
forated to  admit  the  screw, 
and  within  it  is  a  groove  into 
which  the  thread  of  the  screw 
fits  closely. 

The  handle  D,  which  projects  from  the  nut,  is  a  lever 
which  works  the  nut  upon  the  screw.  The  power  of  the  screw 
depends  on  the  distance  between  the  threads.  The  closer  the 
threads  of  the  screw,  the  greater  will  be  the  power ;  but  then 
the  number  of  revolutions  made  by  the  handle  D,  will  also 
be  proportionably  increased ;  so  that  we  return  to  the  general 
principle — what  is  gained  in  power  is  lost  in  time.  The  power 
of  the  screw  may  also  be  increased  by  lengthening  the  lever 
attached  to  the  nut. 

The  screw  is  used  for  compression,  and  to  raise  heavy 
weights.  It  is  used  in  cider  and  wine-presses,  in  coining, 
and  for  a  variety  of  other  purposes. 

Quest. — 365.  Of  how  many  parts  is  the  screw  composed  ?  Describe 
the  screw.  What  is  the  thread  ?  What  the  nut  ?  What  is  the  handle 
used  for  ?    To  what  uses  is  the  screw  appUed  ? 


PROMISCUOUS  QUESTIONS.  361 


PROMISCUOUS  QUESTIONS. 

1.  Two  persons  have  put  in  trade  each  a  certain  sum; 
that  which  the  first  contributed  is  to  that  of  the  second  as  11 
to  15:  the  first  put  in  $1359:  what  did  the  second  con- 
tribute ? 

2.  Twelve  workmen  working  12  hours  a  day  have  made 
in  12  days  12  pieces  of  cloth,  each  piece  75  yards  long. 
How  many  pieces  of  the  same  stuff  would  have  been  made, 
each  piece  25  yards  long,  if  there  had  been  7  more  work- 
men ? 

3.  A  workman  earns  $18,50  by  working  12  days  in  14, 
during  these  14  days  he  spends  50  cents  a  day  for  his  board 
and  gives  4  cents  a  day  to  the  poor ;  on  Sunday  he  triples 
the  alms.  How  long  will  it  take  him  at  this  rate  to  pay  his 
rent,  which  is  $56,  and  a  debt  of  $11,50? 

4.  How  much  time  would  it  require  to  receive  $80  of  in- 
terest with  a  capital  of  $400,  knowing  that  $600  placed  at 
the  same  rate  would  produce  an  interest  of  $90  every  three 
years  ?* 

5.  If  $100  at  interest  gains  $3  every  nine  months,  what 
capital  would  be  necessary  to  gain  $800  every  two  years  ? 

6.  Four  partners  have  gained  $21175  ;  the  first  is  to  have 
$4250  more  than  the  second;  the  second  $1700  more  than 
the  third  ;  the  third  $1175  more  than  the  fourth:  what  is  the 
share  of  each  ? 

7.  The  sum  of  two  numbers  is  5330,  their  difference 
1999  :  what  are  the  two  numbers  ? 

8.  A  person  was  born  on  the  1st  of  October,  1792,  at  6 
o'clock  in  the  morning;  what  was  his  age  on  the  21st  of 
September,  1839,  at  half  past  4  in  the  afternoon  ? 

9.  A  merchant  bought  80  yards  of  cloth,  then  sold  140 
yards,  after  which  there  remained  to  him  one  half  the  quan- 

16 


362  PROMISCUOUS  QUESTIONS. 

tfty  he  had  in  the  store  before  his  last  purchase  :  what  was 
this  quantity  ? 

10.  Sound  travels  about  1142  feet  in  a  second  If  then 
the  flash  of  a  cannon  be  seen  at  the  moment  it  is  fired,  and 
the  report  heard  45  seconds  after,  what  distance  would  the 
observer  be  from  the  gun  ? 

11.  A  person  having  a  certain  sum  borrowed  #65,50,  and 
then  paid  a  debt  of  $94,90  ;  he  received  $56,75  which  was 
due  him,  and  found  that  he  had  $49,30  after  having  expended 
$9,30.     How  much  had  he  at  first  ? 

12.  A  house  which  was  sold  a  second  time  for  $7180, 
would  have  given  a  profit  of  $420  if  the  second  proprietor 
had  purchased  it  $130  cheaper  than  he  did:  at  what  price 
did  he  purchase  it  ? 

13.  A  person  purchased  78000  quills,  for  half  of  which  he 
gave  $4,50  per  thousand,  and  for  the  rest  87^  cents  per  hun- 
dred; he  sells  them  at  1^  cents  each:  what  is  his  profit 
supposing  he  takes  265  for  his  own  use  ? 

14.  In  order  to  take  a  boat  through  a  lock  from  a  certain 
river  into  a  canal,  as  well  as  to  descend  from  the  canal  into 
the  river,  a  body  of  water  is  necessary  46J  yards  long,  8 
yards  wide,  and  2|  yards  deep.  How  many  cubic  yards  of 
water  will  this  canal  throw  into  the  river  in  a  year,  if  40 
boats  ascend  and  40  descend  each  day  except  Sundays  and 
eight  holidays  ? 

15.  How  many  scholars  are  there  in  a  class,  to  which  if 
11  be  added  the  number  will  be  augmented  one-sixteenth  ? 

16.  A  person  being  asked  the  time,  said,  the  time  past 
noon  is  equal  to  J  of  the  time  past  midnight :  what  was  the 
hour? 

17.  What  number  is  that  which  being  augmented  by  85, 
and  this  sum  divided  by  9,  will  give  25  for  the  quotient  ? 

18.  Three  travellers  have  1377  miles  to  go  before  they 
reach  the  end  of  their  journey ;  the  first  goes  30  miles  a  day 
the  second  27,  and  the  third  24  :  how  many  days  should  one 
set  out  after  another  that  they  may  arrive  together  ? 


PROMISCUOUS  QUESTIONS.  363 

19.  A  company  numbering  sixty-six  shareholders  have 
constructed  a  bridge  whith  cost  $200000 :  what  will  be  the 
gain  of  each  partner  at  the  end  of  22  years,  supposing  that 
6400  persons  pass  each  day,  and  that  each  pays  one  cent 
toll,  the  expense  for  repairs,  &c.,  being  $5  per  year  for  each 
shareholder  1 

20.  The  entire  length  of  the  walls  of  a  fort  is  495  yards, 
their  height  8i  yardsj  and  their  thickness  3  yards:  how 
many  years  has  it  taken  to  construct  them,  each  cubic  yard 
having  cost  16  francs,  and  the  expenses  having  been  20086 
francs  per  year ;  and  what  will  this  sum  amount  to  in  dollars 
and  cents,  at  the  custom  house  value  ? 

21.  One-fifth  of  an  army  was  killed  in  battle,  ^  part  was 
taken  prisoners,  and  ^^  died  by  sickness  :  if  4000  men  were 
left,  how  many  men  did  the  army  at  first  consist  of? 

22.  A  person  delivered  to  another  a  sum  of  money  to  re- 
ceive interest -for  the  same  at  4  per  cent  per  annum.  At  the 
end  of  three  years  he  received,  for  principal  and  interest 
£176  8^.     What  was  the  sum  lent? 

23.  A  snail  in  getting  up  a  pole  20  feet  high,  was  observed 
to  climb  up  8  feet  every  day,  but  to  descend  4  feet  every 
night :  in  what  time  did  he  reach  the  top  of  the  pole  ? 

24.  Four  merchants  A,  B,  C,  and  D,  trade  together ;  A 
clears  £76  4^.  in  6  months,  B  £57  10^.  in  5  months,  C  100 
guineas  in  12  months,  and  D,  with  a  stock  of  200  guineas, 
clears  £78  15^.  in  9  months.     Required  each  man's  stock. 

25.  Three  merchants  traded  together  as  follows  :  A  put 
in  $2500  for  3  months,  B  $1750  for  5  months,  and  C  $2000 
for  2  months  :  C's  gain  was  $147,50.  What  must  A  and  B 
receive  for  their  respective  shares,  and  what  was  the  whole 
gain? 

26.  Three  different  kinds  of  wine  were  mixed  together  m 
such  a  way  that  for  every  3  gallons  of  one  kind  there  were  4 
of  another,  and  7  of  a  third :  what  quantity  of  each  kind 
was  there  in  a  mixture  of  292  gallons  ? 

27.  Divide  £500  among  four  persons,  so  that  when  A  ha 
£^,  B  shall  have  i-,  C  i,  and  D  |. 


364  PROMISCUOUS  QUESTIONS. 

28.  Two  partners  have  invested  in  trade  $1600,  by  which 
they  have  gained  $300 ;  the  gain  and  stock  of  the  second 
amount  to  $1 140.     What  is  the  stock  and  gain  of  each? 

29.  How  many  planks  15  feet  long  and  15  inches  wide 
will  floor  a  barn  60^  feet  long  and  33J  feet  wide  ? 

30.  A  merchant  bought  a  quantity  of  wine  for  $430.  He 
sold  55  quarts  of  it  for  $24,50,  and  gained  5  cents  a  quart : 
how  much  wine  had  he  at  first  ? 

31.  Twenty-five  workmen  have  agreed  to  labor  12  hours  a 
day  for  24  days,  to  pay  an  advance  made  to  them  of  $900 ; 
but  having  lost  each  an  hour  per  day,  five  of  them  engage  to 
fulfil  the  agreement  by  working  12  days :  how  many  hours 
per  day  must  these  labor  ? 

32.  If  a  person  receives  $1  for  |  of  a  day's  work,  how 
much  is  that  a  day  ? 

33.  If  14j^  pieces  of  ribbon  cost  $26,50,  how  much  is 
that  a  piece  ? 

34.  What  number  is  that  of  which  i,  J,  and  ^  added  to- 
gether, will  make  48  ? 

35.  A  landlord  being  asked  how  much  he  received  for  the 
rent  of  his  property,  answered,  after  deducting  9  cents  from 
each  dollar  for  taxes  and  repairs,  there  remains  $3014,30. 
What  was  the  amount  of  his  rents  ? 

36.  A  person  traded  360  yards  of  linen  for  cloth  worth 
$1,62  per  yard:  how  many  yards  of  cloth  has  he  received, 
and  for  how  much  has  he  sold  the  linen  per  yard,  knowing 
that  the  price  of  a  yard  of  cloth  is  equal  to  that  of  2|  yards 
of  linen  ? 

37.  If  165  pounds  of  soap  cost  $16,40,  for  how  much  will 
it  be  necessary  to  sell  390  pounds,  in  order  to  gain  the  price 
of  36  pounds  ? 

38.  What  is  the  height  of  a  wall  which  is  14|-  yards  in 
length,  and  -^^  of  a  yard  in  thickness,  and  which  has  cost 
$406, ^it  having  been  paid  for  at  the  rate  of  $10  per  cubic 
yard  ? 

39.  If  the  tare  of  a  quantity  of  merchandise  is  54/^.  7oz,, 
what  is  the  gross  weight,  the  tare  being  4lb.  in  100  ? 


PROMISCUOUS    QUESTIONS.  365 

40.  At  what  rate  per  cent  will  $1720,75  amount  to 
$2325,86  in  7  years  ? 

41.  In  what  time  will  $2377,50  amount  to  $2852,42,  at  4 
per  cent  per  annum  1 

42.  What  principal  put  at  interest  for  7  years,  at  5  per 
cent  per  annum,  will  amount  to  $2327,89  ? 

43.  What  difference  is  there  between  the  interest  of  $2500 
for  4|-  years,  at  6  per  cent,  and  half  that  sum  for  twice  the 
time,  at  half  the  same  rate  per  cent  1 

44.  If,  when  I  sell  cloth  at  8^.  9J.  per  yard.I  gain  12  per 
cent,  what  will  be  the  gain  per  cent  when  it  is  sold  for  10^. 
6d.  per  yard  ? 

45.  A  tea-dealer  purchased  120Z5.  of  tea,  ^  of  which  he 
sold  at  10^.  6d,  per.  lb. ;  but  the  rest  being  damaged,  he  sold 
it  at  a  loss  of  £3  12^.,  after  which  he  found  he  had  neither 
gained  nor  lost.  What  did  it  cost  him  per  lb.,  and  what  was 
the  damaged  tea  sold  for  ? 

46.  A  piece  of  <;loth  containing  5000  ells  Flemish  was 
sold  for  $21250,  by  which  the  gain  upon  every  yard  was 
equal  to  ^  of  the  prime  cost  of  an  English  ell.  What  was 
the  first  cost  of  the  whole  piece  ? 

47.  A  person  lent  a  certain  sum  at  4  per  cent  per  annum , 
had  this  remained  at  interest  3  years,  he  would  have  received^ 
for  principal  and  interest  $9676,80.  What  was  the  prin- 
cipal ? 

48.  Three  persons  purchased  a  house  for  $9202  ;  the  first 
gave  a  certain  sum ;  the  second  three  times  as  much ;  and 
the  third  one  and  a  half  times  as  much  as  the  two  others 
together :  what  did  each  pay  ? 

49.  A  piece  of  land  of  165  acres  was  cleared  by  two 
companies  of  workmen ;  the  first  numbered  25  men  and  the 
second  22  ;  how  many  acres  did  each  company  clear,  and 
what  did  the  clearing  cost  per  acre,  knowing  that  the  first 
company  received  $86  more  than  the  second  ? 

50.  The  greatest  of  two  numbers  is  15  and  the  sum  of 
their  squares  is  346  :  what  are  the  two  numbers  ? 


366  PROMISCUOUS  QUESTIONS. 

51.  A  water  tub  holds  147  gallons  ;  the  pipe  usually  brings 
in  14  gallons  in  9  minutes :  the  tap  discharges,  at  a  medium, 
40  gallons  in  31  minutes.  Now,  supposing  these  to  be  left 
open,  and  the  water  to  be  turned  on  at  2  o'clock  in  the  morn- 
ing ;  a  servant  at  5  shuts  the  tap,  and  is  solicitous  to  know  in 
what  time  the  tub  will  be  filled  in  case  the  water  continues 
to  flow. 

52.  A  thief  is  escaping  from  an  officer.  He  has  40  miles 
the  start,  and  travels  at  the  rate  of  5  miles  an  hour ;  the  offi- 
cer in  pursuit  travels  at  the  rate  of  7  miles  in  an  hour :  how 
far  must  he  travel  before  he  overtakes  the  thief  ? 

53.  Five  merchants  were  in  partnership  for  four  years ; 
the  first  put  in  $60,  then,  5  months  after,  $800,  and  at  length 
$1500,  4  months  before  the  end  of  the  partnership;  the  sec- 
ond put  in  at  first  $600,  and  6  months  after  $1800  ;  the  third 
put  in  $400,  and  every  six  months  after  he  added  $500 ;  the 
fourth  did  not  contribute  till  8  months  after  the  commence- 
ment of  the  partnership ;  he  then  put  in  $900,  and  repeated 
this  sum  every  6  months  ;  the  fifth  put  in  no  capital,  but  kept 
the  accounts,  for  which  the  others  agreed  to  pay  him  $1,25 
a  day.  What  is  each  one's  share  of  the  gain,  which  was 
$20000 1 

54.  A  traveller  leaves  New  Haven  at  8  o'clock  on  Mon- 
day morning,  and  walks  towards  Albany  at  the  rate  of  3 
miles  an  hour  ;  another  traveller  sets  out  from  Albany  at  4 
o'clock  on  the  same  evening,  and  walks  towards  New  Haven 
at  the  rate  of  4  miles  an  hour :  now  supposing  the  distance 
to  be  130  miles,  where  on  the  road  will  they  meet  ? 

55.  An  employer  has  45  workmen,  by  each  of  whom  he 
gains  15  cents  a  day:  .how  long  a  time  would  it  require  for 
them  to  gain  him  $468,93,  and  what  must  he  pay  them  during 
this  time,  he  paying  each  $1,25  a  day? 

56.  When  it  is  12  o'clock  at  New  York,  what  is  the  hour 
at  London,  New  York  being  75°  of  longitude  west  of  Lon- 
don? 

Since  the  circumference  of  the  earth  is  supposed  to  be 
divided  into  360  degrees  (Art.  40),  and  since  the  sun  appa- 


PRCMISCUOUS    QUESTIONS. 


567 


rently  passes  through  these  360^  every  twenty-four  hours,  it 
follows  that  in  a  single  hour  it  will  pass  through  one  twenty- 
fourth  of  360°,  or  15°.     Hence,  there  are 

15°  of  motion  in  1  hour  of  time,  '    . 

1°  of  motion  in  4  minutes 
y  of  motion  in  4  seconds. 

If  two  places,  therefore,  have  different  longitudes,  they 
will  have  different  times,  and  the  difference  of  time  will  be 
one  hour  for  every  15°  of  longitude,  or  4  minutes  for  each 
degree,  and  4  seconds  for  each  minute.  It  must  be  observed 
that  the  place  which  is  most  easterly  will  have  the  time  first, 
because  the  sun  travels  from  east  to  west. 

To  return  then  to  our  question.  The  difference  of  longi- 
tude between  London  and  New  York  being  75°,  the  differ- 
ence of  time  will  be  found  in  minutes 
by  multiplying  75°  by  4,  giving  300 
minutes,  or  5  'hours.  Now  since 
New  York  is  west  of  London,  the 
time  will  be  later  in  London ;  that 
is,  when  it  is  twelve  o'clock  at  New 
York,  it  will  be  5,  p.  m.  in  London ;  or  when  it  is  12  at  Lon- 
don, it  will  be  7,  a.  m.  at  New  York. 

57.  Boston  is  6°  40^  east  longitude  from  the  city  of  Wash- 
ington :  when  it  is  6  o'clock  p.  m.  at  Washington,  what  is  the 
hour  at  Boston  ? 

The  6  degrees  being  mul- 
tiplied by  4  give  24  minutes 
of  time,  and  the  40  minutes 
being  multiplied  by  4  give 
160  seconds,  or  2  minutes 
40  seconds.      The   sum  is 


OPERATION. 

75° 

4_ 

60)300 
Ans.     5  hours. 


OPERATION. 

6x4  =  24m. 
40  X  4  =  160^^=  2m.  40sec, 


26?n.  40sec. 


Ans.  26m.  40sec.  past  6. 


26m.  40sec.,  and  since  Boston  is  east  of  Washington  the  time 
is  later  at  Boston. 

58.  The  difference  of  longitude  of  two  places  is  85°  20"^: 
what  is  the  difference  of  time  ? 


368 


(  PAGES  43,  44.  ) 


ANSWERS. 


REDUCTION. 


3. 
4. 

5. 
6. 

7. 

8. 


10. 

11. 

12. 
13. 
14. 


Ans, 
i     1155;?. 

(  55440/ar. 

52405/ar. 

37245^/.  d. 


b726two  d. 

i  2imtr.  d, 
\  8301(^. 

\Z\2six  d. 

216cr. 
432A/:  cr. 
1080^. 
2160520:  cZ. 
12960^. 
,51840/ar. 

1493^. 
'    5975^r.  <?. 
\  7l700far, 

(  2880(?. 
}  240^. 
(  £12. 


99o-w.  4^.  4J. 
£105. 


15. 

16. 

17. 

18. 

20. 
21. 
22. 

23. 

24. 
25. 

26. 
27, 
28. 
29. 
30. 
31 


Ans, 
,  24]0cr. 
>  £602  10;?. 

(  25920^. 
}    5184cr. 
f  £1296. 


14oz, 

i  184800^r. 
(    13200^r. 

26215^r. 

122Z^>.  2oz.  I8pwt,  9gr 


J 


3005 
24003 
]       72003 
Ll44000^r. 

157ib  75  43. 

86962^r. 

30Ib  45  33  23  7gr. 


26S80lb, 

14T. 

42292Z3. 

20T.  13citJi.l^r.22ZJ.4o;:r. 


(pages  44,  45,  50,  51.  ) 


369 


Ex. 
32. 

33. 

34. 
35. 
36. 

37. 

38. 

39. 
40. 

41. 

42. 

43. 

44. 
45. 
46. 
47. 


Ans. 

403TA9cwt.lqr.24lb. 
Qoz.  231654496(^r. 

5024na. 

m4yds. 
78E.  E.  \qr. 

J  1197^.  E. 

\  23940wa. 
i  9996y(?. 
\  7996^.  E,  \yd. 
(  6664E.  Fr. 

i      3768fur. 
}  I50720rd. 


'      88000y(Z. 
264000/^ 
3168000m. 
_9564000&ar. 
200613ft.  6in. 
C  47 55801  eOObar. 
}  U72qrs.  4bu.  \gal. 
(  3184Sar.  over. 

12374P. 

2214262;?5'./J5.  72sq.in. 
2800P. 


Ex. 

48. 

49. 
50. 
51. 
52. 
53. 
54. 
55. 
5Q. 
57. 
58. 
59. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
68. 

69. 

70. 


Ans. 
93A.  2R.  16P. 

818ilf.  \62A.  3R.  23P. 

\0840S.ft. 

3760128>S.  in. 

440cords. 

43742cords  32  S.  ft. 


13  tuns 

37800p?. 

970lhf.  ank 

85248^2, 


32832jo2f. 

2972l6hfpt. 

1800^aZ. 

^     1408p^. 


I240sacks. 

315576005ec. 

189733554^60. 

(240yr.  I0da.4hr.  28m: 
(  38sec. 


ADDITION. 


4. 
5. 

6. 

7. 
8. 


787676921. 

10570011. 

15371781930. 

45105211. 


16* 


9. 
10. 
11. 
12. 
13. 


6001001250561. 

6000037684799. 

128738075326 

21890459447. 


370 


(  PAGES   51 — 57.  ) 


Ex, 

Ans, 

Ex. 

Ans. 

14. 

1819857171437. 

47. 

403A.  IR.  IP. 

15. 

$108,892. 

48. 

16. 

$1057,87. 

49. 

3209tun.  Ohhd.  27gal, 

17. 

$800,076. 

50. 

5422pun.  57gal.  2qt, 

18. 

51. 

A60tier.  29 gal.  \qt. 

19. 

$5498,043. 

52. 

297gal.  2qt. 

20. 

$67476,840. 

53. 

21. 

.6684  5s.  Id, 

54. 

323bar.  Ifir,  4gal, 

22. 

£205  35.  \Qd. 

55. 

.      3150AM.  16^aZ.  \qt. 

23. 

56. 

5220hhd.  Agal.  2qt. 

24. 

£240  Qs.  8J^. 

57. 

,528L.ch.  I3bu.  2pk, 

25. 

2^Q2lb,  \oz,  \Qpwt, 

58. 

26. 

344Z5.  \oz.  \9pwt,  20gr. 

59. 

3842qr.  6bu.  2pk. 

27. 

4:621b.  9oz.  Upwt, 

60. 

409SCOWS  l2L.ch.  I9bu. 

28. 

61. 

4299yr.  7^^^mo.  2wk. 

29. 

511flj  115  33. 

62. 

525/710.  Ov^k.  4da. 

30. 

294HJ  Of  73. 

63. 

31. 

''36*  55  63  13  I8gr. 

64. 

4444hr,  23m.  50sec, 

32. 

464fij  05  53. 

33. 

34. 

3030cwj^  Iqr,  27lb, 

APPLICATIONS. 

35. 
36. 
37. 

92cwt,  2qr,  I5lb.  lOoz, 
3471b,  loz,  Ur, 
41\yd.  2qr.  \na. 

1. 
2. 

1605260acres. 
i  1st  3  j/rs.  42390529^. 
;►  last     "        4530902^. 

38. 
39. 

3S21E,  Fr, 

3. 
4. 

$2051423,77. 

40. 
41. 

4768jS.  Fl  Oqr,  2na, 
489X.  Imi,  6fur. 

5. 

(  15995942  coins, 
i  $5668663 z=zvalue. 

42. 
43. 

4487/wr.  35r^.  5t/d, 

6. 

{  Imports,  $303955539. 
I  Exports,  $287820350. 

44. 

eUft,  Oin,  Ibar, 

7. 

1287462. 

45. 

509A,  2R,  18P. 

8. 

3617900. 

46. 

4797A.2R.  UP. 

9. 

(  PAGES  57 — 64.  ) 


371 


Ex. 

Ans, 

Ex, 

Ans. 

(  29884  to  Br,  N.  Amer, 

17.                                1104087. 

10. 
11. 

}  667Z0  to  U.  S. 
(96654  entire  number. 
l3S500tons. 

18. 

In  1790,     3924829. 
1800,     5305941. 
1810,     7265579. 

12. 

2'6\ll\men. 

1820,     9638191. 

13. 
14. 

$70560. 

r  681  No.  of  vessels. 
}  403  sail  vessels. 

«■ 

1830,   12861192. 
1840,   17063350. 

r  In  1790,     607897 

(144  steam  vessels. 

1800,     893041. 

15.. 

r$  977911  of  gold. 
}     1567420  of  silver. 

19. 

1810,   1191359. 
1820,   1627428. 

(    2545331  entire  sum. 

1830,   1998318, 

16. 

,       1840,  2487355. 

SUBTRii 

lCTION. 

1. 

.     81328. 

19.                          79Ri  105  63. 

2. 

7559. 

20. 

3. 

£7  18^.  9f^. 

21.                          133  03  15^r. 

4. 

33891899020240993. 

22.                             8flJ  105  73. 

5. 

23.                   12 T.  llcwt.  3qr, 

6. 

499972609093220149. 

24.                     2cwt,  2qr.  26lb. 

7. 

149299788316514071. 

25. 

8. 

$179,577. 

26.                134Z^>.  l4oz.  ISdr. 

9. 

$79,324. 

21.                  134yJ^.  2qr.  3na. 

10. 

28.               124^.  E.  3qr.  3na, 

11. 

$999,955. 

29.               96E.  Fr,  2qr,  Ina, 

12. 

$107,576. 

30. 

13. 

$566,034. 

31.                     171,.  2mi.  6fur, 

14. 

$985,997. 

32.                              34rJ.  Ayd. 

15. 

33.                      Ard,  3\yd.  2ft, 

16. 

9oz.  \lpwt,  20gr, 

34.                       3ft.  Oin,  Ibar, 

17. 

I5lb.  Soz,  I6pwt. 

35.                                       

18. 

2oz,  ISpwt,  2\gr. 

36. 

37A.  2R.  34P. 

.372 


(  PAGES  64—68.  ) 


Ex, 
37. 

38. 

39. 

40. 

41. 

42. 

43. 

44. 

45. 

46. 

47. 

48. 

49. 

50. 


61.     <[ 


62. 
63. 

64, 

65. 
66. 


Ans. 
lA,  IR,  26P. 

4A.  2R.  39P. 

7tun  2hhd.  bbsal. 


Itier. 

Igal.  3qt, 

7gaL  2qt.  Ipt. 

\har. 

SJlr, 

Igal. 

lOlhar. 

Ifir. 

Agal. 

eshhd. 

2gal 

.  3^?. 

27L.  ch 

Obu 

\pk. 

2weys  4qr. 

2hu. 

52qr 

6bu, 

3pk, 

Ex.  Ans. 

51.  2yr.  llyf^mo.  3wk. 

52.  l27mo,  3wk.  Ma. 

53.  147c?Gf.  2\hr.  b^min. 

54.  52 Ar.  50mm.  b4sec. 

PROMISCUOUS    EXAMPLES. 


55. 

£3  9^. 

56. 

£121  \7s.  0\d. 

57. 

£980  2s.  U 

58. 

180T.  llcwt.  I2lb 

59. 


60. 


(  5yr.  ll-Jj^2^o.  2wk. 
I  6da.  9hr.  22min. 

340  3y—dif.  oflat. 
I650  18^=  "     "  lona 


Newton's  age  was  84yr.  2mo.  26da. 

Euler's        "       "     76yr.  4mo.  22da. 

Lagrange's        **     77yr.  2m.o.  Wda. 

Laplace's    "       "     78yr.  Ada. 

From  Newton's  death  to  Jan.  1st, 

1846,  was 

"     Euler's         "         "         " 

"     Lagrange's  "         "         " 

"     Laplace's      "         ''         " 


3279hhd. 

,    9372lddiff. 

'  2428921  pop.  of  state. 

(  13277872  diff. 

I  18535786  whole  pop. 


$92449341,16. 


67. 
68.' 
69. 
70. 
71. 
72. 


118yr.  9mo.  \2da. 
62yr.  3mo.  24da. 
32yr.  8mo.  21  da. 
IByr.  9mo.  bda. 

*  £5742078. 

$52315291. 

$2458211. 


$49282,03. 
$3466051,78. 


(  PAGES  68 — 75.  ) 


373 


Ex, 


73. 


Ans, 

'From  the  founding  of  St.  Augustine,  280yr.    6mo.    9da 
"       "         "  Jamestown,       238yr.  lOmo.    Ada, 

"     Battle  of  Princeton,  Q9yr,    2mo.  I4da, 

"  Surrender  of  Cornwallis,  64yr.  4mo.  29da, 
A^^ashington's  Inauguration,  56yr.  lOmo.  17 da, 
Washington's  Death,  46yr.    3mo,    3da, 

the  French  Berlin  Decree,  39yr.    3mo.  26da, 

"     Orders  in  Council,  28yr,    Amo^    6da. 

"     Declaration  of  War,  33yr.     8mo.  29da, 

.  "     Capture  of  the  Guerriere,       33yr.    Qmo.  29da, 

"  "        "    "    Macedonian,  33yr.    4mo.  2Sda, 

u  York,  32yr.  \0mo.  20da, 

"  "        "  Fort  George,         32yr.    9mo.  21  da, 

"     Defeat  at  Sackett's  Harbor,  32yr.    9mo.  20da. 

"    Battle  of  Lake  Erie,  32yr.    6mo.    Ida, 

"         ''      of  Chippewa,  31yr.     Smo.  12 da, 

"        "      of  Niagara,  31yr.    7mo.23da. 

"     Sortie  from  Fort  Erie,  3lyr,    6mo. 

"    Battle  of  New  Orleans,         31yr.    2mo,    9da, 

"    Death  of  Adams  and  Jeffer- 
son, \9yr.    8mo,  I3da, 

"     Compromise  Bill,  13yr.    Imo.    5da. 

"    Death  of  Lafayette,  12yr.     9mo,  28da, 

"    Removal  of  the  Cherokees,     7yr.    9mo,  22da, 


MULTIPLICATION. 


L 

2. 

3. 

4. 

6. 

7. 

8. 
23. 
24. 
25. 
26. 


6776368. 

9. 

68653214. 

10. 

4563272. 

11. 

1301922. 

19. 

20. 

•  556321146764. 

21. 

1747125213301. 

22. 

2324684880333. 
71109696492112. 


129359360000. 
13729103000000. 
664763206000000. 
8799238229600000. 
2526426017908695000000. 
1093689368445084378777040. 

8371562339213807802080112 


374 


(  PAGES  75—79.  ) 


Ex. 

27. 

Ans. 
72058988008174745973090826 

28. 

95666032459647278072171264. 

29. 

4896. 

23. 

£2  Os.  6d. 

30. 

234048. 

24. 

£2  Is.  8^d. 

31. 

25. 

32. 

314986464. 

• 

26. 

27. 

£2  I9s.  ed. 
£8  3s.  9d 

Art.  67. 

28. 

£16  8s.  7^d. 

1. 

(  $70,840. 
{     85,008. 
(     99,176. 

29. 
30. 
31. 

IIZ^. 

£22  13^. 
6oz.  9pwt.  I2gr. 

2. 

$7834,14. 

32. 

197 yd.  Iqr.  Ina. 

3. 

$12517,68. 

33. 

£3l9s.  ^d. 

4. 

$77079,456. 

34. 

£66  I9s.  6d. 

5. 

35. 

6. 

$341,25. 

36. 

£65  I9s.  9^d. 

7. 

$98,94. 

37. 

£208  13^.  9d. 

8. 

$813,020. 

38. 

£154  12^.  3d. 

9. 

$5869,75. 

39. 

£42  Is.  6d. 

10. 

40. 

11. 

$2426,15. 

41. 

£819  6s. 

12. 

$15169,50. 

42. 

edoib. 

Soz.  I8pwt.  I6gr. 

13. 

$162,25. 

43. 

75A.  3R.  39P. 

14. 

$21935,214. 

45. 

£19  10^.  8ld. 

15. 

46. 

16. 

$963,66. 

47. 

£33  3^.  If^. 

17. 

$18844,01. 

48. 

•   £83  2^.  8d. 

18. 

£81  65.  10^. 

49. 

£137  7s.  3d. 

19. 

2^T.7cwL  3qr,27lb.  8oz. 

50. 

£698  2s. 

20. 

51. 

21. 

£llOs,  2d. 

52. 

222cwt.  I81b. 

22. 

£1  9s.  2d. 

53. 

I5cwt 

.27lh.  lloz.  7dr. 

(  PAGES  79—88.  ) 


375 


Ex 
55. 

Ans. 
£2687  I8s.  3d. 

Ex. 
65. 

Ans. 
£566  5^.  3ld. 

56. 

£351  2^.  l^d. 

66. 

£17038  105.  lie?.  2^ar. 

57. 



67. 

£12422  2s.  Id.  2f/ar. 

58. 

£62  Is.  7^d. 

68. 

£2875  05.  7\d. 

59. 
60. 

£15299  185.  4d. 
£51  7^.  2^d. 

69. 
70. 

£658  05.  llt^.  \\far. 

61. 

£344  0^.  6d. 

71. 

£50  125.  U. 

62 

. 

72. 

£2  105.  2d.  2|/ar, 

DIVK 

3I0N 

2. 

407294^^11 . 

9. 
10. 
11. 
12. 
13. 
14. 

131809655ifff|^ 

3. 
4. 
5. 
6. 

13195133i|||. 
125139201i|?f^ 

9948157977^VVWy. 
59085714j-Yx. 

14243757748|f^il. 

1258127|fJ||. 
123456789. 

7. 

15395919iffA|. 

15. 

8. 

30001000/yW^. 

16. 

119191753t^^W6- 

17. 

9001 

84442401 8274624226||f Iff. 

Art.  79. 

Art.  81. 

1. 

132. 

3. 

17085f|-. 

2. 

4871000. 

4. 

6763921. 

3. 

4. 

5. 
6. 

•  6129^V 
3095^^4. 

7128368. 

5 

918546. 

8. 

Art.  80. 

9^ 

5203802^^^. 

2. 

387. 

10. 

118D5558|f. 

3. 

133. 

11. 

39096821x^2T- 

4. 

201. 

12. 

34297219^^. 

376 


(  PAGES  88—93.  ) 


Ex. 

Ans. 

Ex. 

Ans. 

13. 

11. 

2E.  E.  3qr.  if  Jna. 

14. 

10823637JI. 

12. 

£1  17^.  6d. 

15 

650f|f. 

13. 

ed.  l-^{^far. 

16.206190192477|A3|g4  69o. 

14. 

lis.  llfJ. 

15. 

Art.  82. 

IG. 

£40  10^.  6d. 

2. 

156557943V_7_5_. 

17. 

Is.  9d 

3. 

18. 

lO^d. 

4. 

16871651^VtVo- 

19. 

4s.  9^^-^d. 

5 

474625653\Vo%. 

20. 

%j  t 

6. 

8905748eiM^. 

21. 

lOoz.  Ibpwt.  UjWgr. 

7. 
8. 

1421076222^V4'^5^- 

22. 

{  \lcwt.  2qr.  Wlb.  lAoz, 
I  l3^\dT. 

39E.  Fl.  Iqr.  Sna.  l^in. 

9. 

4087692937,3_B_i_7_i_5_. 

23. 

10. 

7943859f§. 

11. 

119092l|fi§. 

APPLICATIONS. 

12. 

71400714374. 

13. 

1. 

21  T.  lOcw;^.  2qr.  6|ZZ>. 

14. 

19245303W^Vo- 

2. 

3. 

$73,296  +  . 

4. 

$0,0899+. 

EXAMPLES    IN    DENOMINATE 

NUMBERS. 

5. 

$1,283  +  . 

6. 

20-37868  re?n. 

2. 

£15  19^.  M.  l^far. 

7. 

3. 

£\  Us.bd.  l|/ar. 

8. 

(  $1,255  +  . 
I  $1,244  + 

4. 

155.  Id.  3i{far. 

5. 

9. 

$25141072,267  +  . 

6. 

£9  I8s.  3d.  3^-f^far. 

10. 

$0,06  +  . 

7. 

9yd.  2qr.  \^na. 

11. 

$166743,259  +  . 

\    4A.  32P.  20^^.  yd. 
'  (Isq.ft.  72sq.in. 

12. 

8. 

13. 

37-2588  rem. 

9. 

Sib.  Ipwt.  \^^Qgr. 

14. 

38-190  rem 

10. 

15. 

$334477,744  +  . 

(  PAGES  93—108.  ) 


377 


PROPERTIES    OF    THE    9's. 


Ex. 
1. 
2. 
3. 


Art.  85. 

Ans, 
2.     7. 

Ex. 
3. 

1.     7. 

2. 

Art.  86. 

140487982-7. 

3. 

Ans. 
177234105-3. 


Art.  87- 

, 503602-7. 

'  Excess  in  minuend^  7. 

45991735-7. 
Excess  in  minuend.  8. 


VULGAR  FRACTIONS. 


Art.  98. 


1. 

2. 
3. 
4. 
5. 
6. 
7. 
8. 

1. 
2. 
3. 

4. 
5. 
6. 
7. 
8. 


2. 
3. 


Art.  99. 


Art. 

100. 

35 

1. 

67 

8    • 

28^ 

!i- 

2. 

34 

T83- 

S¥- 

3. 

ilh- 

132 

4. 

17   • 

5. 

i-hh- 

w- 

6. 

\m- 

1_2_6 

7. 

327 

Tl  • 

18  23- 

W- 

8. 

Art. 

101. 

^h' 

tV- 

1. 

2. 

^h- 

3. 

II,  ft  ¥• 

_2  1_ 

4, 

iAL 

147. 

456* 

7     » 

2     • 

A^6- 

5. 

ifX. 

30, 

6. 

T47' 

7. 

w. 

246 

21   • 

9 

8. 

449 

449 

4  92' 

T2» 

-T-- 

Art.  102. 

:5i   6  8.   10  2.   119   153   255  289 

5  7*  7  6,  114'  TTS",  T7T,  2^5^,  T2~3* 

111.  JL5.6.   ^0.8.   9JL_   6  5  143 

153'  204»  272,  Tl9,  85,  1%J* 


S78 


(  PAGES   109—118.) 
Art.  103. 


Ex. 
3. 

4. 

30       20      15 
9  0^»     6  0'     4  5» 

\h 

Fo» 

Arts, 

kh  1^.  tV.  I>  i 

A»  T5»  T%»  9»  h  h 

GREATEST    COMMON 

DIVISOR. 

1. 
2. 
3. 
4. 
5. 
6. 

4. 
45. 

630. 
267. 

2. 
3. 

2d    METHOD, 

Art.  105. 

43,  3,  5 
3,3,2,2,2,5, 

12. 

Art.  106. 

7. 

8. 

2. 

3 

8. 

4. 

3. 

25 

9. 

3. 

4. 

10. 

, 

5. 

15 

least    COMMON    MULTIPLE. 


3. 

4. 
5. 
6. 

7. 

8. 

9. 

10. 


840. 

11. 

147. 

2d 

METHOD. 

840. 

3. 

4. 
5. 

1260. 

7200. 
2520. 

78. 

84. 
1  nnft 

6. 

7. 
8. 

1008. 

iUUO. 

156. 

10800. 

REDUCTION    OF    VULGAR    FRACTIONS. 


Art.  110. 

6. 

2. 

12f. 

7. 

3. 

nyd. 

8. 

4. 

b^hu. 

9. 

5. 

10. 

(  O   77        94      71700 
)  -^T25'    ^^»     '694T> 

V  ^^72301' 


219|. 
14. 


(pages  118—126.  ) 


Ex, 

'1. 


Ans, 
10731^V 


Art.  111. 


3. 
4. 

5. 

6, 

7. 

8. 

9. 
lO. 
11. 
12. 
13. 

2. 
3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 

2. 
3. 


287 
6     • 
34_51_3      .7  89  9.      69047 

51       J  9      J        10  0~' 

3.8J  7  7 
^       104"' 

IjVlP-l    ill  7.2     59267822 
^  V5       »       104     »  8 "7  9  • 


28278 

15  1     • 

1346 

9       • 

3_7219 

9  9       • 

17  49  3  8 3049 

9  9  9^9         • 


16106 
9        • 

V7-I  =  78. 


Art.     12. 


Ex. 
4. 

5. 

6. 

7. 

2. 
3. 

4. 

3. 
4. 
5. 

2. 
3. 
4. 
5. 
.6. 


Art.  113. 


h¥s- 

7. 

^• 

2- 

4. 

J. 

5. 

6. 

4 

y 

7. 

m. 

8. 

41 

sr- 

9. 

69 

^49' 

10. 

183 

11. 

5381' 

12. 

if^. 

^IF. 

13. 

379 


Ans. 

332497 
18  1~  • 


^-'■iP- 

«_22_5_8. 

Art.  114. 

1__5 

7    • 

Art.  115. 

w/ 

-3.7  5  1 
10504* 

rHih- 
=  8|H  • 

Art.  116. 

m- 

_99 
332- 

345f. 

42l>?6"' 
2  8  0  9  8 5  2  0 
63  6  8  99  ey* 


Art.  117, 


52. 5.      1080       2  2  2  0  0 
6  00  J      6  00  '    "6^0  0~"' 
2_Q_P       6  2.      15.5  0 
50  »    5  0»        5  0"^' 
IJLSP       2  48.      9  0.0 
144  J    Ti4'     144* 
J-U.      1JL2_6       15  7  5 
T2  6»      1  26  J      T2¥  • 


-5    9 

28 »  2""8' 
^0_6_6  4   1_4  9_7_6   165  7  6 
4  03  2  >   4  03  2  '   4  0  3^2  » 
2G712 
4032  • 
12  6   140   3  0   105 
210»  210»  2T0»  2T0^^ 
C  2  3  04   2.2A0   15  12 
;4032'  4032'  4  03  2» 
)   3528   7_6_6_0  8 
V.  4032'   4032^  • 


880 


(  PAGES  126—132.  ) 


Ex. 

Ans, 

^a?. 

14. 

3. 

Art.  118. 

4. 

3. 

T2'   T2»   A- 

5. 

4. 

!!'  V'  !!'■ 

6. 

5. 

5  4»    2  4»     24' 

7. 

6. 

V¥,  W,  ¥/. 

7. 

8. 

Art.  119. 

9. 

2. 

If,  H,  .^. 

10. 

Arts. 

122   5J.   4  4 

8  »   8  ,   8  • 

7  2_   60   320 
5~60,  360»  360* 

6  7    18   3  0  0 
T2  0,  T2  0»  12  0' 


iAH   624 
14  4,  14  4, 


1_2_00   2_0_7_9 
144 ,   144  • 


6    8    9    10 
T2,  T2,  1:2,  T2' 

3  6   6  0   5_0   6_3 
90»  90'  90'  90* 

16   36   40   42   33   34 

48,  48,  48,  48,  48,  48* 


REDUCTION  OF  DENOMINATE  FRACTIONS. 


Art.  121 


3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 

2. 
3. 
4. 
5. 


Art.  122. 


6. 

7. 

4^^hhd. 

8. 

%U7- 

9. 

^Ts^o- 

10. 

zh^hd. 

11. 

12. 

AVW^.^ 

13. 

^l^cwt. 

14. 

Iledo'^' 

15. 

£f. 

16. 

17. 

^It^*. 

18. 

Ti5^^. 

19. 

^^hhd. 

2, 

64Ib, 

3. 

32d, 

4. 

480m. 

5. 

^i^P. 

6. 

259_2-00^eC. 

5040^^. 
Ipwt, 

jib. 


2448^^ 


Art.  123. 

9oz.  \2pwt, 

2R,  20P. 

3s,  4d. 

b2gal,  2qt. 


(  PAGES  132—136.  ) 


381 


Ex. 

Ans» 

Ex. 

Ans. 

7. 

24gal  2qt, 

9. 

TiHok-o^-^' 

8. 

10. 

-N^da, 

9. 

7oz,  4pwt, 

r    ^Vo     of    2da. 

10. 

Ipi.  Ihhd.  Slgal  2qt. 

xlh   of    3da. 

11. 

I    loz.  Apwt. 

11. 

xlio   of    4cfa. 

12. 

Imi.  Qfur.  \Qrd. 

,-|l^   oflOJa. 

13. 

12. 

-T^VoOf25(fa. 

14. 

Byd,  \qr,  \\na. 

15. 

Ipi.  Ihhd.  Igal. 

13. 

If^- 

16. 

29gal.  Iqt.  lj\pt. 

14. 

\Uwt. 

17. 

98da,  8hr.  Am.  36if  ^ec. 

15. 

T\Wf>' 

18. 

16. 

liE.E. 

19. 

5^.  Ad. 

17. 

20. 

dcwt,  3qr.  6lb. 

18. 

^lll 

21. 

lOS.ft.2l6S.in. 

19. 

\groat. 

22. 

^Ogal.  2jm§jqL 

20. 

^quarter. 

21. 

z\yd- 

Art.  124. 

22. 

3. 

23. 

sfkft- 

4. 

j\hhd. 

24. 

^hhd. 

5. 

j^-^CWt. 

25. 

^hs.yd. 

6. 

^^hhd. 

26. 

rhhL.ch. 

7 

79 

27. 

•  • 

B-3360^*- 

8. 

28. 

UUhd. 

ADDITION    OF    VULGAR    FRACTIONS. 


2. 
3. 
4. 
5. 


Art.  126. 


£2|. 


6. 

2. 
3. 
4. 


Art.  127. 


60 
IT* 


£453 

i0  194 
9  4  5~" 

10^0  8 


382 


(  PAGES  137—141.  ) 


Ex. 

Ans, 

Ex, 

Ans, 

5. 

4. 

2qr.  17lb.  loz,  S^ld, 

6. 

Hi^- 

5. 

Imi.  Sfur,  ISrd. 

7. 

"mi- 

7. 

Art.  128. 

8. 

IcwL  Iqr.  27lb,  l3oz. 

2. 
3. 

4. 

9. 
10. 
11. 
12. 
13. 
14. 

^        lu.ejfj. 

£3  125. 
2oz.  lOpwt.  \2gr. 

170AV 

841-1. 

5. 
6. 

7. 

2E.  E.  4qr.  Of  na. 
Sfur.  25rd.  3yd.  l-^^in. 

8. 
9. 

6  16 
fi783 

"8  8  0- 

15. 

16. 

^2R.  20P.  I  i  Sq.ft. 

3hhd.  37gal  3}(]t. 

Art.  129. 

17. 
18. 

2. 

U^in.  = 

=  llly^. 

2da.  2hr.  12m. 

3. 

2da 

L  14iAr. 

19. 

55Ja.  lOAr.  Q\m, 

SUBTRACTION  OF  VULGAR  FRACTIONS. 


o 

Art.  131. 

m- 

10. 
11. 

75f. 

3. 

mh 

12. 

36^^. 

4. 

13. 

47f. 

2. 

Art.  132. 

h- 
^  H- 

14. 
15. 
16. 

3. 

4. 

H- 

Art. 

133. 

5. 

A- 

2. 

lUr. 

59771. 

b^sec, 

6. 

3. 

lib.  8oz 

.  l^pwt.  IQgr. 

7. 

/o. 

4. 

legal.  2qt.  Ipt 

^^\gi' 

8. 

ii. 

5. 

9s.  3d. 

9. 

62^^9^0- 

6. 

(  PAGES   141—146.  ) 


Ex.  Ans. 

7.  4cwt,  Iqr,  Iblh.  \oz.  9|Jr. 

8.  14^.  3fcZ. 

9.  ^oz.  Ipwt.  \2gr. 
10.             ^Tcwt,  Iqr.  27lb.  Soz. 


883 

Ans. 


Ex. 

11.  

12.  \mi.  \fur.  \Qrd 

13.  202da.  2\hr.  45m.  ^2\sec 

,  .         (  Troy  oz.  \pwt.  IS^gr. 
\  greater. 


MULTIPLICATION    OF    VULGAR    FRACTIONS. 


Art.  135. 


2. 
3. 

4, 
5. 
6. 
7. 
8. 
9. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


^h- 


33|A. 

42i. 

124lfi. 


1 19 « 9 6^9 


Art.  136. 


8^- 


20. 

9  8 


540. 


13. 

3. 

4. 
5. 
6. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


408  7  9  6T' 


Art.  137. 


608tV 
5987|-- 


10622f. 


GENERAL  EXAMPLES. 


1157 
■^308' 
303 
700* 
140 
8T8T- 


$5i. 

$36. 

$7lf. 

£2  35.  9^. 


¥6. 


DIVISION    OF    VULGAR    FRACTIONS. 


Art.  139. 


2 
3  7- 

^285' 


4. 

5. 

6. 


379 

T9Tro5* 

rVa- 


384 

(  PAGES    1 

Ex, 

Ans. 

f^ 

61 

1  < 

77  7- 

Art.  140. 

1. 

h 

2. 

29^. 

3. 
4. 

mm- 

5. 

153801. 

6. 

30HH- 

7. 

I 

8. 
9. 

71AV 

10. 

20fg. 

U. 

H- 

12. 

9A. 

13. 

14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 


Ans, 

KQ31758 
°°3334T« 

1    7  8  03 
21. 
T^96* 


82^0^^. 

4f\cts, 
$5,25. 
$2j-V5. 


95j\cts, 

$0  454 


DECIMAL  FRACTIONS. 


Art.  143. 

15. 

1. 

2. 
3. 

41.3. 

16.000003. 

5.09. 

16. 

17. 

.000003. 
.00039. 

4. 

65.015. 

Art.  144. 

5. 

6. 

7. 

•   2.00000003. 
.492. 

1. 
2. 

$17,389. 
$92,895. 

8. 

3000.0021. 

3. 

9. 

47.0021. 

4. 

$47.25. 

10. 

5. 

$39,397. 

11. 

39.640. 

6. 

$12,003. 

12. 

300.000840. 

7. 

$147.04. 

13. 

.650. 

8- 

14 

50000.04. 

9. 

$4,006. 

(  PAGES    L 

53— 

Ex. 

Ans, 

Ex. 

10. 

$14,039. 

13. 

11. 

$149,332. 

14. 

12. 

$1328.005. 

15. 

159.  ) 


385 

Ans. 

$0,058. 
$3856.02. 


ADDITION    OF    DECIMAL    FRACTIONS. 


1. 

1306.1805. 

11 

2. 

528.697893. 

12 

3. 

13 

4. 

1.5415. 

14 

5. 

446.0924. 

15 

6. 

27.2087. 

16 

7. 

88.76257. 

17 

8. 

18 

9. 

1835.599. 

19 

10. 

397.547. 

31.02464. 
1.110129. 


$641,249. 

.111. 

4.0006. 

2.413009. 


$1132.365. 


SUBTRACTION    OF    DECIMAL   FRACTIONS. 


2. 

3277.9121. 

13. 

17.949 

3. 

249.60401. 

14. 

,699993 

4. 

9.888899. 

15. 

5. 

16. 

.999 

6. 

2.7696. 

17. 

6373.9 

7. 

1571.85. 

18. 

365.007495 

8. 

.6946. 

19. 

20.9942 

9. 

.89575. 

20. 

10. 

21. 

10.030181 

11. 

1379.25922. 

22. 

2.0294 

12. 

99.706. 

2. 
3. 


MULTIPLICATION    OF    DECIMAL    FRACTIONS. 


329.307391. 
742.036196. 


4. 
5. 


26.99178 


17 


386 


(  PAGES  159—164.  ) 


Ex. 

Ans. 

Ex. 

Ans. 

8. 

10376.283913. 

14. 

7. 

275539.5065. 

15.                  933.8253150762. 

8. 

.020621125. 

16. 

.25. 

9. 
10. 

17. 
18. 

.0025. 
.00715248. 

175.26788356. 

11. 

.00043204577. 

19. 

12, 

215.67436625. 

20. 

.02860992. 

13. 

.000000000294. 

21. 

2.435141056. 

CONTRACTION    IN 

MULTIPLICATION. 

2. 

258.13005. 

4. 

3. 

162.525. 

5. 

3566163. 

DIVISION    OF    DEC 

IMAL    FRACTIONS 

. 

Art.  152. 

r  254.7347748. 

2. 
3. 

2.22. 

8.522. 

10. 

25473.47748. 
254734.7748. 
2547347.748. 

4. 

33.331. 

25473477.48. 

5. 

,254734774.8. 

6. 

12420.5. 

11. 

r  25.05068. 

12. 

1918+. 

7. 

250.5068. 

2505.068. 

25050.68. 
^250506.8. 
r  48.65961. 

13. 

14. 
15. 
16. 

.00473  +  . 

1.74412. 
69.7125. 

8. 

' 

4865.961. 
48659.61. 
486596.1. 

17. 
18. 

12976  +  . 
.0049589+. 

^4865961. 
'41.622. 

Art. 

154. 

416.22. 

2. 

10970. 

^. 

-( 

4162.2. 
41622. 

3. 
4. 

60200. 

416220. 

4162200. 

5. 

100. 

(  PAGES   164—170.  ) 


887 


Eoi. 

Ans, 

Ex. 

Ans, 

[10. 

4. 

$11055.2925 

100. 

1000. 

30. 

5. 
6. 

$140,625. 

6. 

< 

20. 

7. 

$20.87. 

2000. 

8. 

$3731.123. 

* 

12. 
1200. 

9. 

224.58.5.  yd. 

Art.  165. 

• 

,500000. 

10. 
11. 
12. 

$365.61525. 
$1.35 

3. 

8.3111  +  . 

269  acres  =  area. 

4. 
5. 

1.563  + . 

13.     < 

$13573.204  =  cost. 
$50,458+  =z  average 

Art.  156. 

^     price. 

'      $7631.8855  =  cZ(/e;?« 

1. 
2. 

339.51 300  ly(/. 
155.1011/5. 

14.     - 

son^s  share. 
$5723.914125  =  share 
of  each  of  the  other 

3. 

$88,141. 

^     sons. 

CONTRACTION    IN    DIVISION. 


2. 
3. 


2. 
3. 
4. 
5. 


7. 
8. 
9. 


\- 


4. 

35.2843-3  rem. 

5. 

3TI0N    OF    VULGAR    ] 

''RAC 

.125.  .0159+. 

11. 

12. 

.5.  .0028  +  . 

13. 

1.496+. 

14. 

1.333  +  .  .1629+. 
792  +  .  4.666+. 

15. 

.02343  +  . 

16. 

17. 

.0003. 

18. 

.2224  +  . 

19 

11.5834036625-6  reTTi 
3202.8870-1  rem. 


.000000488+. 
.8571  +  . 


.2571  +  . 

.8947+. 

.008033  +  . 

.23903  +  . 

1.5555  +  . 


388 


(  PAGES   170—174.  ) 


Ex, 

Ans, 

Ex. 

Ans. 

20. 

.15909+. 

23. 

21. 

$100.8. 

24. 

1.25. 

22. 

$17.85. 

25. 

3.0339+. 

REDUCTION    OF    DENOMINATE    DECIMALS. 

Art.  160. 

7. 

10.16666  +  . 

1. 

.0546875Z6. 

8. 

£.3729+. 

2. 

£.325. 

9. 

3. 

10. 

£.^325757  +  . 

4. 

.029166Ja.  +  . 

11. 

.12968/5.  +  . 

5. 

3.9375j9>t. 

12. 

.05oz. 

6. 

.375cZa. 

13. 

J  .3987( 
)  Arith., 

}3lb.-{-troy.  (See 

Art.  20,  p.  23.) 

7. 
8. 

71.15l7ni.  +  . 

14. 

V                           ' 

9. 

.00396W.  +  . 

15. 
16. 

.633928cw;^.  +  . 
S  .042965c«7^.  +  .        (See 

10. 

l.byd. 

\\  Arith., 

Art.^0,  p.  23.) 

11. 

.66251b, 

17. 

.3125yd. 

12. 

.7282yr.  +  . 

18. 

.55E.E. 

13. 

19. 

14. 

£25.977  +  . 

20. 

.48125J. 

15. 

.9375cwt. 

21. 

.00992MJ.  +  . 

16. 

.7391m. +  . 

22. 

.104166c/^.  +  . 

17. 
18. 

.2325  T. 

23. 
24. 

.07472yr.  +  . 

19. 

.7129(^a.  +  . 

25. 

.26175^. 

Art.  161. 

26. 

.1005113^11.  +  . 

1> 

£19.8635+. 

Art.  162. 

2. 

£2.325. 

1. 

2qr.  Ulb. 

3. 

.6255. 

2. 

2qt.  \pt. 

4. 

3. 

5. 

2.5yd. 

4. 

20gaL  Iqt. 

6. 

I M687  5lb. 

5. 

\36da,  2lhr. 

(pages  174—181.) 


389 


Ex. 
6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 


3. 
4. 
5. 
6. 

7. 
8. 
9. 


Ans, 
\qr.  l4oz.  5dr,-{-. 


2mi,  2Ard.  byd.  lOm.-f . 

Qs.  9d. 

Qcwt.  3qr, 

8P. 


Ex.  Ans. 

14.  I2dr. 

15.  208<Za.  2hr.  23m.  33sec.  +  . 


16. 
17. 
18. 

19. 

20. 


£2  Is,  lOd.i- 
£5  I2s.  9^d.  +  . 


J  Icoom  I  strike  2pk. 
\  3qt.  ljpt.-\-. 

15  134 


CIRCULATING  OR  REPEATING  DECIMALS. 
Art.  164. 
Factors  of  den.  5x5x2.  decimal  value  .06. 
37X5X2.   .08^8  + 

2x2x2x2x2x2x2x2x5.   .01328125. 

Art.  165. 

5x5x5x2.   .028. 

5X5X5X5.   .0176. 

2X2X2X2X2X2X2.   .1328125. 


REDUCTION    OF    CIRCULATING    DECIMALS. 


Art.  175. 


3. 

4. 

riT»  TT>  y 

Art.  176. 

4. 

5. 

/       163          41 
^    16500'     90' 

1. 


2. 


Art.   177.     Sec,  2. 

^2.4^181818^. 
). 5  925925^+. 
(  .008^497133"+. 
Sec,  4. 

r  165^.16  416416'+. 
)  .04'b40404V.. 
(  .03>77777'  +  . 

.5'333333V. 

[  A  757575'-^. 
L  7^577577  -4-. 


390 


Ex. 
2. 


3, 

4. 


(  PAGES  184—187.  ) 
Art.  178. 


Ans, 
.1875. 


.0  0344827586206896551724137931^-1- „ 


ADDITION    OF    CIRCULATING    DECIMALS. 


2.  95.2"^82964/-f. 

3.  69.74"203112^+. 

4.  55.6"209780437503'  +  . 


5. 
6. 


47.4  754481'  +  . 


SUBTRACTION    OF    CIRCULATING    DECIMALS. 


2. 

45.7  755  -f. 

6. 

3. 

2.9"957'  +  . 

7. 

4.619"525'+. 

4. 

5.09. 

8. 

1.0923  7  +  . 

5. 

.65''37001 6280906'+. 

9. 

1.3462^937"+. 

MULTIPLICATION    OF    CIRCULATING    DECIMALS. 


2. 

7.. 

3. 

1.098"086'+. 

8.* 

11.068735402'+. 

4. 

1.641  !>  +  . 

9. 

.81654"l68350'+. 

5. 

1.7183^39'+. 

10. 

189.301977'+ 

6. 

1.4710\)37'+. 

2. 
3. 
4. 
5. 


DIVISION    OF    CIRCULATING    DECIMALS. 


13.570413  961038  +. 


7.71954  +. 
26.7837"42857l''  +  .  i    9. 


6.  ^  3.1 45+. 

7.  3^82352941 17647058%. 


15.48  423  +. 


(paces  188—199.  ) 


891 


RATIO  AND  PROPORTION  OF  NUMBERS. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


Art.  183. 


Arts. 


Art.  185. 


1 
i 

2* 

1 
3- 


1 

To- 


3.  5..J-.   1.  }f. 


Ex, 
1. 
2. 
3. 

4. 
5. 
6. 

7. 
8. 


1. 
2. 
3. 


Art.  187. 

Ans, 

9 

:     8  : 

:     18  :     16. 

16 

:     9  :  , 

48  :     27. 

13 

:    19  : 

•     52  :     76. 

16 

:  21   : 

.     80  :   105. 

35 

:  42  : 

210  :  252. 

23 

:  45  : 

:  207  :  405, 

Art.  188. 


6. 

f 
A- 


OF    CANCELLING. 


3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 


Art.  190. 


11. 

9. 
2J. 

8. 

12. 
6. 


1. 
2. 
3. 
4. 
5. 

1. 
2. 
3. 


Art.  191. 


Art.  192. 
2:8: 
2  :  7  : 


63 

27. 


323. 

1  fx369 


4. 

21. 


RULE  OF  THREE. 


applications. 

3. 

1. 

$330. 

4. 

2. 

£9  Qs,  8J.. 

5. 

$2762,50. 
3300i^. 


392 

Ex 
6. 

7. 
8, 
9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

27. 

28. 
29. 
30. 
31. 

32. 

33. 
34. 

35. 


(pages  199—204.  ) 


3hr,  2771. 


Ans, 
32  men, 

49||^ec. 


16432 J  miles, 

$121,875. 


20  days. 

4200bu, 

£253  lOs.  ^d. 


6oz. 


\5^%dr. 


£8  I6s.  2^^%d, 


Ex, 
17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

25. 

26. 


Ans, 
£1913  6s.  8d. 

6^oz, 

£1270  Is,  QiiJ. 


£39637  10^. 

120  yards, 

190  guineas, 

Idj^da. 


108000. 


,  Rate  3mi,  Ifur,  IS^jvd,  per  hour. 


Time  20^r.  2lm.  2^sec. 


17  times  round. 
4^1  days. 


ISoz.  per  day. 
i  588000Z6.  total  weight. 
!    42000ZZ>.  spoiled. 

2018E.  Fl.  2qr. 

£9  3s.  9J. 

3b\yd.  baize. 

'  cost  11^.  2c?. 

per  yard. 


I'tIt/^^ 


36. 
37. 
38. 

39. 

40. 
41. 


8^.  2d.  3||ff l/ar. 

$112,86. 

(  whole  weight,  588000/5. 
I  they  received,  546000/6. 

\  whole  weight  94O8OOO02:. 
\  l4oz  per  day. 


RULE    OF    THREE    BY    ANALYSIS. 


2. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 


504  miles. 
$2,08. 


87  miles.  11. 
12. 
13. 
14. 
15. 
$380.  16. 
1,429  +  .  17. 
30  days,i  18. 


105  days. 


lOdyd.  2ft. 
51  days, 
27  days. 


lib.  5oz. 


9|J- 


64*  bottles. 


(  PAGES  204—208.  ) 


393 


Ex.  Ans. 

19.  25yr.  202da.  2hr. 

20.  He  gained  $246,75. 


Ex, 
21. 
22. 


Ans, 


RULE    OF    THREE    BY    CANCELLING 


2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


8^.  8c?. 


160  days, 
27001b. 


60  days. 

23352bu, 

£7  4s. 


20  days. 


13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 

22. 


54  days, 

6  days, 

12  c/ay^. 


3600. 

8|  yards. 

16  months, 

121  yards. 


jC41,  D's  part. 
'  £61  10^.  joaz'J  Z>y  cacA 
o/'^Ae  others. 


EXAMPLES    INVOLVING    FRACTIONS. 


2. 
3. 

4. 
5. 

6. 

7. 

8. 

9. 
10. 
11. 
12 

24. 


£1  I8s.  6d, 

£682  I8s.  9d. 

£112  12^.  9^od. 


£102  7^.  7d.  +  . 

$1,431  iV 
f  14.581. 
14  days. 


.005172  guineas.  + . 
.7l428cwt.-\-. 


13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 


$52.50. 
$638.08  + 


$21  10^.  Hd, 

62.734375  days. 

£77  3^.  7^d, 

$5,625. 


'  At  the  equator, 
"  Madras,    dsc. 
"  Madrid,       " 
"  Petersburg" 

17* 


2700. 

293.28^aZ5. 

lU^^hrs, 

1038f|f  miles, 

lOlO^^  miles^ 

794||J  miles, 

519i|^  miles. 


894 


(  PAGES  213—219.  ) 


DOUBLE  RULE  OF  THREE. 


Ex. 

Ans. 

Ex. 

5 

13. 

6. 

116765W. 

14. 

7. 

36  days. 

15. 

8. 

2808qrs. 

16. 

9. 

168. 

17. 

10. 

18. 

11. 

10  days. 

19. 

12. 

9600  men. 
PRAC 

20. 
TICK 

13. 

£91  1 5.  ^^%d. 

32. 

14. 

£44  9^.  8\^d. 

33. 

15. 

£38  3.9.  3f  c/. 

34. 

16. 

£19  \s.  3jf^. 

35. 

17. 

36. 

18. 

$1,575. 

37. 

19. 

$547,50. 

38. 

20. 

$5. 

39. 

21. 

$108. 

40. 

22. 

41. 

23. 

£10  155.  |f|(f. 

42. 

24. 

£18  75.  6j\d. 

43. 

25. 

£3  7s.5Ud. 

44. 

26. 

£2051  I3s.  10//g(Z. 

45. 

27. 

46. 

28. 

$1617. 

47. 

29. 

$643,75. 

48. 

30. 

$519,75. 

49. 

31. 

£1877  105.  9yW- 

540^.  yd. 


Ans. 

14:00. 

36yd.  long. 


27  men. 

£11  23296^. 

4da.  llhr.  54^^^^7n. 

8571f. 


£348  145.  l^d. 
$135,375. 

$15. 
$1854. 


£199  l5.  lO^d. 


$8,10. 


£550  II5.  lO^d. 

£661  17 s.  3j%d. 

£445  5s.  4-^^\d 

$65,90625 


$148,2890625. 
$48,91796875. 


(  PAGES    220— 

224. 

)                         39.5 

TARE  AND  TRET. 

Ex. 
1. 

2. 

65cwt,  Iqr.  I9lb, 
4cwt,  Iqr.  Iblb.  Soz, 

Ex. 
6. 

7. 
g 

Arts. 

$808,71. 

£912  14^.  ^d. 

3. 
4. 
5. 

$265,16. 
£59  4;?.  3/g^. 

9. 
10. 

1 

ST.  Scwt,  3qr.  5lb. 
6T,  12cwt.  3qr.  3^lb. 
Value,$30G,7246875. 

11. 

12. 
13. 
14. 

15. 
16. 
17. 


(  In  leaf,    $26,0586+  per  cwt, 
I  In  rolls,   $29,5504+     "     " 

Net  weight,   I6lcwt.  Iqr.  8.96llb.-\-. 

Value,  $1177,709  +  . 


(  Net  weight,  27cm;^.  2qr.  S^Ib. 
I  Value,  $232,837+. 

(  Net  weight,  lOcwt.  27^lb, 
Rvalue,  $4,0417  +  . 

(  Net  weight,  20cm;^.  2qr.  1 3}lb, 
}  358.3 12^«/.+ 

(  Freight,  $444,306+. 

^  Net  weight,  298cw;^.  2qr.  23,454/6.+ 

(  Freight,  $355,464+. 


Ex. 

1. 

2. 
3. 
4. 
5. 
6. 
7. 


Art.  204. 


PERCENTAGE. 

Ex. 

Ans. 


432bar. 

42hhd. 

$10,80. 

$24,25. 


205  boxes. 


1. 
2. 
3. 
4. 
5. 


Ans. 
742gal  3qt.  3lgi. 

Art.  205. 

55    "      " 


20  per  cent 


16|" 


396 


(  PAGES  227—234.  ) 


SIMPLE  INTEREST. 


Art.  207. 

Ex, 

^   Ans, 

Ex. 

Arts. 

10. 

$31,928125. 

4. 

■  11. 

$121,77275 

5. 

$803,25. 

12. 

6. 

$450,32760. 

13. 

$609,45776. 

7. 

$4853,844. 

Art.  212. 

8. 

$643,83375. 

2. 

$16474,3855  +  . 

9. 



3. 

$1449,70998. 

10. 

$235,764. 

4. 

$371,86875. 

11. 

$1205,9208. 

5. 

12. 

$1375,8144. 

6. 

$266,277. 

13. 

$12959,584. 

7. 

$14096,5  +  . 

14* 

8. 

$4479,618 

Art.  208. 

9. 

$1149,552 

3. 

$1225,511. 

10. 

4. 

$44,20845. 

11. 

$40900,9335. 

5. 

$6015,4272. 

12. 

$15955,65195. 

6. 

$357,9165. 

Art.  213. 

7. 

3. 

$511,5357. 

8. 

$270,5175. 

4. 

$167,30. 

9. 

$2953,22685. 

5. 

10. 

$7765,4.504. 

6. 

$555,465. 

11. 

$1578,6625. 

7. 

$4200. 

12. 

8. 

$2643,8386. 

Art.  210. 

9. 

$7,00. 

3. 

^  $3,2727875. 
\\  $145,1545. 

10. 

11. 

$587,6311  +  . 

4. 

$67,278. 

12. 

$3974,5187  +  . 

5. 

$5,56529  +  . 

13. 

$3329,1468  +  . 

6. 

$0,4314  +  . 

14. 

$1137,0592  +  . 

7. 

15. 

8. 

$1,13559. 

16. 

$678,2599  +  . 

P. 

$13,68. 

17. 

$4824,2366. 

(  PAGES  234—245.  ) 


397 


Art.  214. 


Ex. 
3. 

Ans, 
$43,2049  +  . 

Ex.  . 

4. 

$8,1855. 

5. 

'Inter 

est  at  4  per  cent, 

5  "     " 
51    «      u 

6  "     " 

u             y       ((       (t 
71     u       u 

c         8     "     " 

SI  '*     « 

<         9     "     " 

$45,837. 

$57,29625. 

$63,025875. 

$6^,7555. 

$80,21475. 

$85,944375. 

$91,674. 

$97,40:^625. 

$"103,13325. 

6. 

4. 

7. 
8. 
9. 

$380,28952. 

$669,7096875. 

$25571,2473  +  . 

5. 
6. 

$4640,r7326. 
$1976,6305  +  . 

Art.  217. 

A 

2. 
3. 

RT.  215. 

£45  8^.   \ld. 

2. 
3. 

4. 

$5360,545  +  . 
$8922,927  +  . 

4. 
5. 
6. 

7.             J 
8. 

£A^  lOs.  2c/.  + 

£662  3i\  + 

£216   Is.  \0\d.-\- 

£219  18^.  0\d.-\- 

2. 
3. 
4. 

Art.  220. 

$3976,848  +  . 

$575,569  +  . 

$1424,682  +  . 

9. 

A 
1. 
2. 
3 

£6 
RT.  21 

$: 

8  5s,  M.-\- 

6. 

394,3256  +  . 

$697,986. 

$3339,6+. 

5. 
2. 
2. 

Art.'221. 

5  per  rent. 
Art.  222. 

'iyr.  Ctmo. 

REDUCTION  OF  CURRENCIES. 


3.  £1073  18;?.  \\d. 

4.  $1967,892  +  . 


5. 
6. 


$2551,733  +  . 


398 


(  PAGES  247—256.  ) 
COMPOUND  INTEREST. 


Art    223. 

Art.  224. 

2. 

$57,3048+. 

2. 

$578,740  +  . 

3. 

$73,0154-. 

3. 

$8611,128+. 

4. 

$41,216  +  . 

4. 
5. 

$7058,617+. 

5. 

6. 

$2647,996  +  . 

6. 

'       $48165,938  +  . 

7. 

£11  ISs.  lld.  +  . 

7. 

$14523,553  +  .! 

8. 

$9974,685  +  . 

LOSS  AN 

D  GA 

[N. 

Art.  225. 

6. 

3. 

Loss  of  $0,75. 

7.. 

$337,50. 

4 

$0,966  +  . 

8. 

$217,50. 

9. 

$4,108. 

Art.  227. 

10. 

•     25  per  cent. 

1. 

11. 

2. 

12  per  cent. 

12. 

C  Whole  gain,  $13,00. 
I  Gain,  20  per  cent. 

.3. 

$1,25. 

4. 

$1,20.    25  per  cent. 

13. 

$2,05. 

5. 

$6,33i. 

14. 

$1,0311. 

-      COMMISSION  A 

ND  BROKERAGE. 

2. 

$49725. 

11. 

$46260. 

4. 
5. 

12. 

$23700. 

$7235,142  +  . 

13. 

$25420,195. 

6. 

$7816,091  +  . 

14. 
15. 

$965,30. 

7. 

15  tons. 

16. 

$1995. 

8. 

$213500. 

17. 

$13573,56. 

9. 

18. 

^  149T.  IScwt.  2qr.  \2lh. 
I  eoz.  l-^dr. 

^0. 

$59110. 

(  PAGES  260—268.  ) 


399 


BANK  DISCOUNT. 

Art.  239. 

Art.  240. 

Ex, 
1. 

Ans, 

Ex. 
2. 

$344,59  +  . 

2. 
3. 

4. 

$15240,54. 

$5,840  +  . 

83393,504. 

3. 

4. 
5. 
6. 

$5734,32  +  . 
$695,64  +  . 
$118,85  +  . 

5. 

$29,0097+. 

7. 

$1740,60  +  . 

6. 

8. 

$1057,51  + 

DISCOUNT. 

2. 

$1551,918  +  . 

9. 

3. 

je33  17^.  7|rf.  +  . 

10. 

$3869,407-+  . 

4. 
5. 

■  je223  5^.  8i.  +  . 

11. 
12. 

\hu.  2-f^qts, 
$2109,236  +  . 

6. 

$5620,175  +  . 

13. 

$2763,694  +  . 

7. 

$702,485  +  . 

14. 

8. 

£804  19^.  5c/.  +  . 

15. 

He  lost  $6,473+. 

INSURANCE. 

2. 

$5168,59. 

5. 

* 

3. 

(  $237,60. 
I  $158,40. 

6. 

7. 

$504. 
$39,375. 

4. 

" 

■$252. 
$126. 

$84. 

$56. 

.    $42. 

8. 

9. 
10. 
11. 

$306,25. 
$450. 

$18,75. 

2. 


ASSESSING  TAXES. 
.4685+  per  cent,  \    3. 


$37901125. 


400 


(  PAGES  270—276.  ) 


EQUATION  OF  PAYMENTS. 


Ex. 

Ans. 

Ex. 

-An^. 

2. 

12  months. 

9. 

3. 

10. 

6mo.  ^days 

4. 

9  months. 

11. 

7 mo,  3da 

5. 

68^  (Zay^. 

12. 

Jan.  25th. 

6. 

8|  months. 

13. 

8mo.  lOda 

7. 

67 A  c?ay5. 

14. 

PARTNERSHIP  OR  FELLOWSHIP. 


J    (  A's  share,  $1714,285-1-, 
'•  I  B's  share,    $285,714  +  . 

(A's  share,  £4030. 

B's      '*  £3980. 

C's      "  £3980. 

D's     "  £4010. 


A's  share,  $5000. 

B's      "       $2500. 

4.  <!  C's      "       $3'333,33  +  . 

D's      "       $2500. 

LE's      "       $6666,67  +  . 


DOUBLE  FELLOWSHIP. 


2. 


A's  share,  £9  12^. 
B's      "       £14  8s. 


GENERAL     EXAMPLES     IN     FEL- 
LOWSHIP. 

75  cents  on  the  dollar, 
A's  part,  $375,27J. 
B's     "     $171. 
C's     "     $968,42^. 
LD's     "     $532,05. 

'A  paid      $3000. 

B     ''        $3000. 

J  C     "        $9000. 

'*•  I  A's  gain,  $250. 

B's     "     $250. 

.C's     "      $750. 


4. 
5. 
6. 


'  1st  son's  share, 

$3333,331. 

2d         "         $3000. 

3d         "         $3000. 
.4th        "         $2666,66|. 

'  $750  widow's  gain. 
$375  younger  son's  gain 
$3000  widow's  share, 

.$1500  younger  son's  do. 


'A's  loss,  $46,526  +  . 

B's     "     $130,273  +  . 

C's    '*     $238,213  +  . 
.D's    "     $334,983  +  . 

{  A's  share,  $36. 
)  B's      « 


(  PAGES  277—292.  ) 


401 


Ex. 
2. 

3. 


ALLIGATION  MEDIAL. 


Ans. 
$0,84375. 

$0,28ff. 


Ex. 

4. 

5. 


Ans, 
730. 


ALLIGATION  ALTERNATE. 


Art.  252. 

(  2lb.  at  Sets. 
<  2lb,  at  \Octs. 
(  Qlh.  at  14c^^. 

'  3  parts  of  1 6  carats. 

2  "      of  18      " 

3  "      of  23      " 
.5     "      of  24      " 

iQgal,  at  10^. 
3gaL  at  14^. 
4gal.  at  21^. 
SgaL  at  245. 

Art.  253. 


2. 


{\2lu.  of  oats. 
12^?/.  of  barley. 
\2hu.  of  rye. 
9Qbu.  of  wheat. 

32gaL  of  spirits. 
32gal.  of  Eng.  brandy. 
40firaL  of  French  do. 


Art. 


254. 

29lb.  at  55 

at  6s. 

at  85. 

^29       at  95. 

{45gal.  at  45 
5gal.  at  65. 
5^a/.  at  85. 
bgal.  at  IO5. 


CUSTOM  HOUSE  BUSINESS. 


1. 

82812,5. 

4. 

$1442,875 

2. 

$418,068. 

5. 

3. 

$251,45  +  . 

1. 

2. 
3. 


TONNAGE  OF  VESSELS. 


225t\  ^(?w5. 

438.59  tons-\'. 

729gV_  ^0;,^. 


4. 
5. 


300.14  ^onj+. 


402 


(  PAGES  295—316.  ) 


GAUGING. 

Ex, 
2. 

Art.  267. 

Ans. 
32.4938m. 

Ex, 
2. 

3. 

4. 

Ans. 
162.613  beer  gaL'\', 

3. 

28.1010m. 
Art.  268. 

147.384  M?m6^a/.  +  . 

1. 

197.459  w;me^aL  +  .. 

LIFE  INSURANCE. 

- 

2. 

$144.  1    3. 

$189,55. 

ENDOWMENTS  AND  ANNUITIES. 

1. 

$228,11  +  . 

2. 

1. 

2. 
3. 

4. 


1. 

2. 
3. 
4. 


Art.  293. 

$8591,975. 

$8637,168  +  . 
•    $9777,636. 

Art.  294. 

$5630,065. 


EXCHANGE. 
5. 


£14014  lQs.2d-\-. 
$6005,368  +  . 


',873  +  . 

Art.  295.   - 
7  per  cent*  above  par. 


2. 

3.  

4.  8i 597  francs  66  centimes: 

Art.  296. 
1.  $6657,693. 

5  $1250,52,  3  per  cent* 


(    nearly  below  par. 


DUODECIMALS. 


Art.  301. 


15/i.  5^ 


2ft.  6' 3''  IV. 
5ft.  10^  7''. 


examples  in  addition  and 
subtraction. 


1. 

2. 
3. 

4. 


5ft.  8'  2''  V'\ 
\5ft.  4'  10^^  4'''. 


2lft.  6''  5''  5' 


The  percentage  is  estimated  upon  the  custom  house  value. 


(  PAGES  316—330.  ) 


Ex. 
5. 

6. 


3. 

4. 

1. 
2. 
3. 

4. 
5. 
6, 
7. 
8. 


36/^.  4^  6''  b''\ 
22ft.  2'  V  W'. 

Art.  303. 

214/^  V  V  6''\ 


Ex. 
5. 

6. 

7. 

8. 

9. 

10. 


INVOLUTION. 


1953125. 

343. 

3600. 


1889568. 
1. 
1 

4* 

.001. 


9. 
10. 
11. 
12. 
13. 
14. 
15. 


403 

Ans. 
232ft.  2'  8'' 

seeft.  8'  3''. 

2  cords  5  cord  feet. 
57ft.  4'  6^^ 


185/^.  6'  4''  3^' 


.0001. 
37.4544. 
1000000. 


666024768837. 


EXTRACTION  OF  THE  SQUARE  ROOT. 


3. 
4. 
5. 
6. 

7. 

2. 
3. 
4. 
5. 


2 

3 


Art.  309. 


462. 
1506.23  +  . 
3897.89  +  . 


4698. 


Art.  310. 


57.19  +  . 

69.247  +  . 

2.091 +. 


Art.  311. 


.0321. 
2.104. 
6.906. 


EXTRACTION  OF  THE  CUBE  ROOT. 


Art.  313. 


179. 


4. 
5. 
6. 


.5236  +  . 

.4203  +  . 
1.0682  + 
.86602  +  . 
.93309  +  . 

319. 
439 
638. 


404 


(  PAGES  330—339.  ) 


Ex. 

7. 
8. 

1. 
2. 
3. 
4. 
5. 
6. 


Ans, 


Art.  314. 


3002. 

.5032  +  . 
.955. 
2.35. 


.707. 
1.505. 


Ex. 

7. 


1. 
2. 
3. 
4. 
5. 
6. 
7. 


Art.  315. 


-4n^. 
3.026. 


f 


3 
S* 

.829  +  . 
.822  +  . 


ARITHMETICAL  PROGRESSION. 


2. 
3. 


2. 
3. 

2. 
3. 


Art.  318. 


Art.  319. 


Art.  320. 


$1,53. 
$205. 


4  years. 

5  miles. 


]  Last  term  34. 
'  Sum         162. 


4. 

1. 
2. 
3. 
4. 


5  miles  1300  yards. 

GENERAL    EXAMPLES.  . 

89. 
4. 


$4. 

3  miles  each  day. 
100     "     in  all. 

3. 

$272. 


2. 
3. 

4. 


GEOMETRICAL  PROGRESSION. 


Art.  322. 


J&25600. 
$61,44. 


Art.  323. 


3.  $196,83. 
4. 


£204  15^. 
$295,24. 


Art.  326. 


MENSURATION. 

3. 

36  acres. 


5A.  IB.  15P. 
135  J. 


(  PAGES  341—350.  ) 


405 


Efe. 

^4w5. 

Ux. 

Arts, 

Art.  329. 

3. 

4071.5136. 

1. 

437A  2i?.  34P.  +  . 

4. 

196996571.7221045^.7715. 

2. 
8. 
4. 
6. 
6. 

2'91A  2E.  16P. 

35A  Oi?.  25P. 

20^. 

40^. 

15A. 

2. 
3. 
4. 

Art.  338. 

268.0832. 

2144.6656. 

259992792079.869 -h. 

1. 

24A.  IE,  8P. 

5. 

904.78085^. /if. 

S. 

13008052'.  yds. 

Art.  340. 

Art.  331. 

1. 

91005^./^. 

2. 

21A  OE.  39.824P. 

2. 

1440  sq.ft. 

8. 

921sq.fL  10'  6''. 

Art.  341. 

4, 
6. 

.    704.1 25sg.  yG?5. 
60A  SE.  12.8P. 
270A  li?.  24P. 

2. 
3. 

4. 

1105925oZw^  in. 

42lsolidft. 

Art.  332. 

5. 

lSS20solidft. 

2. 

.     584.3376. 

Art.  343. 

3. 

4. 

125.664. 

2. 

233.SSS +sq.  ft. 

3. 

2S2l.4:4sq.  in. 

Art.  333. 

4. 

628S.2sq.ft. 

2. 

7418. 

Art.  344. 

8. 

4360.835  +  . 

2. 

36442.56. 

Art.  334. 

3. 
4. 

13571.712. 
9650.9952. 

2. 

19.635. 

5. 

7363.125. 

8. 

153.9384. 

4. 

1.069016+. 

Art.  346. 

2. 

4380. 

Art.  336. 

3. 

2484. 

2. 

615.7536. 

4. 

5620. 

406 

(  PAGES    350— 

-362. 

) 

Ex, 

Ans. 

Ex. 

Ans, 

5. 

5760. 

Art.  349. 

6. 

14400. 

1. 

241.86/1 

7. 

1800. 
Art.  348. 

2. 
3. 

lY.204/i^. 
14.142/iJ. 

Art.  350. 

2. 

9160.9056. 

1. 

302.9702/«f. 

3. 

8659.035. 

2. 

28yo?. 

4. 

2827.44. 

3. 

77.8875/^. 

MECHA^^CAL  POWERS. 


Art.  355. 

1.  40/6. 

2.  25/6. 

3.  50/6. 

4.  20/6. 

5.  40/6. 

6.  l^w.,  l^m.,  2m.,  4^7^.,  &c. 

7.  64/6. 

8.  150/6. 


Art.  361. 


1. 


60/6. 


2. 

40/6. 

3. 

Art.  362. 

20/6. 

1. 

" 

^ft- 

2. 

Art.  363. 

HA 

1. 

40/6. 

2. 

100/6. 

2. 

60/6. 

PROMISCUOUS  QUESTIONS. 


1. 

2, 
3. 
4. 
5. 

6. 


$1853,131+. 

5  7  pieces. 

V2wk.  Z^da. 

4  years. 


{4 til  partner'' s  sJiare  $2 500. 
3d         "  "     S3675. 

2d         "  "     $53  75. 

1st         "  "     $9625. 


7. 

8. 

9. 
10. 
11 

12. 


j  Greaternumher,S664l. 

(Less  "         1665^. 

46?/r.  llmo.  20da.  lO^hr. 

120  yards. 


$31,25. 

r  $6760  />r/ce  7ie  would  have 

<       paid. 

(  $6890 p?'iceaotuaIhj  paid. 


(  PAGES  362—365.  ) 


407 


Ex, 
13. 
14. 
15. 

18. 
19. 
20. 
21. 
22. 
23. 

24. 
25. 

26. 

56. 

37. 
38. 
39. 
40. 
41. 

46. 

47. 

4a 

49. 


$454.9375. 
23599680  cubic  yards. 


16. 
17. 


Ans 

3  olclocli, 

140. 


j  The  second  6|  days  after  the  third. 
(  The  first     5^  days  after  the  second. 
,36  +  .  fA's  share, 


7500  men. 

£157  105. 

4  days. 

A's  stock,  £304  165. 
B's      "      £276 
C's      "      £210 


62-|-^aZ.  ofthe  Jstkind. 


83 
146' 


2d 
3d 


27. 


B's 

C's 
ID'S 


£194  165.  lif^. 


28. 

29. 
30. 
31. 
32. 
33. 
34. 
35. 


"  £129  175.  4f^cZ. 

'*  £97  85.  ^^. 

"     £77  185.  b^d. 

Gain  of  1st,  $120, 

"       2d,  $180.' 

108^. 


10  hour  is, 

$1,75. 
$1,853+. 


130|Q  yards  of  cloth. 


12cwt. 


$42,34y2- 

4  yards. 

16/6.  1502. 


Price  of  hnen  per  yard,  $0,5 8^  J. 


4yr.  11  mo.  27J|fG?a. 


42. 
43. 
44. 
45. 


$1724,363+. 

356,25. 

34f  per  cent. 


Cost  per  yard, 
Entire  cost, 


$4,90JiV 
$18378,37fi. 

$8640. 
$920,20  =  what  the  1st  gave. 
$2760,60  =        "         2d      " 
$5521,20  =         "         3d      " 

{$28|-  amount  paid  each  workman. 
1st  company  cleared  87|f-  acres. 
2d         '^  "        77ii      " 

Cost  of  clearing,       $8^9^^  per  acre. 


408 

Ex, 
60. 
51. 
62. 


63. 

64. 
55. 
6'3. 


(pages  3G5— 367.  ) 


Ans, 


6  o'cloclc  3w.  i^W^sec, 
140  miles. 

Share  of  tlie  1st,  $2019,651+. 
"  "       2d,  $4871,803+. 

"  "       3d,  $4815,805+. 

«  "      4tli,$6467,739+. 

"  "       5tli,$1825. 

69f  miles  from  New-Haven. 

5/ir.  41wi.  20sec. 


TFCiC  KFD. 


